Novel techniques for pulsed field gradient NMR measurements

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NOVEL TECHNIQUES FOR
PULSED FIELD GRADIENT NMR MEASUREMENTS

















By

WILLIAM W. BREY


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1994














ACKNOWLEDGMENTS


I would like to thank Janel LeBelle, Igor Friedman, and

Don Sanford for construction of the gradient coil

prototypes, and Jerry Dougherty for performing the

simulations of the coils they all helped to construct.

Stanislav Sagnovski, Eugene Sczezniak, Doug Wilken, and

Randy Duensing participated in many helpful discussions

concerning gradient coils. Debra Neill-Mareci provided the

excellent illustration of a gradient coil in Figure 22. For

their part in the microscopy project, thanks go to Barbara

Beck, Michael Cockman, and Dawei Zhou. Ed Wirth and Louis

Guillette provided the samples. For help measuring eddy

current fields I am grateful to Wenhua Xu, and to Steve Patt

for help with the software. Thanks go to my parents, Mary

Louise and Wallace Brey, and my brother, Paul Brey, for

encouragement and help with red tape. Paige Brey has my

special thanks for her extensive help preparing the thesis.

Katherine Scott, Richard Briggs, Jeff Fitzsimmons, and Neil

Sullivan enriched my graduate experience with their wide

knowledge and diverse interests. Thanks go to them for

their enthusiasm and for reading this thesis. Raymond

Andrew served as supervisory committee chairman. Thanks go

to Thomas Mareci for directing the research, for providing









financial and moral support, and for encouraging me to

pursue this work to its conclusion.


iii















TABLE OF CONTENTS



ACKNOWLEDGMENTS .......................................... ii

ABSTRACT ................................................. v

GENERAL INTRODUCTION ....................................... 1

MEASUREMENT OF EDDY CURRENT FIELDS ....................... 5
Introduction ........................................ 5
Literature Review ................................... 11
Spin-Echo Techniques ................................ 23
Stimulated Echo Techniques .......................... 28
Results ............................................. 34
Conclusion .......................................... 41

GRADIENT COIL DESIGN ..................................... 46
Introduction and Theory ............................. 46
Literature Review ................................... 50
Field Linearity ..................................... 61
Efficiency .......................................... 62
Eddy Currents ....................................... 68
Coil Projects ....................................... 72
Amplifiers ....................................... 72
16 mm Coil for NMR Microscopy .................. 76
9 cm Coil for Small Animals .................... 79
15 cm Coil for Small Animals ................... 87
Concentric Return Path Coil .................... 98

SYSTEM DEVELOPMENT FOR NMR MICROSCOPY .................... 126
Introduction ........................................ 126
Literature Review ................................... 128
Instrument Development .............................. 133
Results ............................................. 149
Conclusion .......................................... 153

CONCLUSION ............................................... 155

REFERENCES ............................................... 156

BIOGRAPHICAL SKETCH ...................................... 162










Abstract of Dissertation Presented to the Graduate School of
the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



NOVEL TECHNIQUES FOR PULSED FIELD GRADIENT NMR MEASUREMENTS

By

William W. Brey

December, 1994


Chairman: E. Raymond Andrew
Major Department: Physics

Pulsed field gradient (PFG) techniques now find

application in multiple quantum filtering and diffusion

experiments as well as in magnetic resonance imaging and

spatially selective spectroscopy. Conventionally, the

gradient fields are produced by azimuthal and longitudinal

currents on the surfaces of one or two cylinders. Using a

series of planar units consisting of azimuthal and radial

current elements spaced along the longitudinal axis, we have

designed gradient coils having linear regions that extend

axially nearly to the ends of the coil and to more than 80%

of the inner radius. These designs locate the current

return paths on a concentric cylinder, so the coils are

called Concentric Return Path (CRP) coils. Coils having

extended linear regions can be made smaller for a given

sample size. Among the advantages that can accrue from

using smaller coils are improved gradient strength and









switching time, reduced eddy currents in the absence of

shielding, and improved use of bore space.

We used an approximation technique to predict the

remaining eddy currents and a time-domain model of coil

performance to simulate the electrical performance of the

CRP coil and several reduced volume coils of more

conventional design. One of the conventional coils was

designed based on the time-domain performance model.

A single-point acquisition technique was developed to

measure the remaining eddy currents of the reduced volume

coils. Adaptive sampling increases the dynamic range of the

measurement. Measuring only the center of the stimulated

echo removes chemical shift and B0 inhomogeneity effects.

The technique was also used to design an inverse filter to

remove the eddy current effects in a larger coil set.

We added pulsed field gradient and imaging capability

to a 7 T commercial spectrometer to perform neuroscience and

embryology research and used it in preliminary studies of

binary liquid mixtures separating near a critical point.

These techniques and coil designs will find application

in research areas ranging from functional imaging to NMR

microscopy.














GENERAL INTRODUCTION


As pulsed field gradient technology for NMR matures,

new and diverse applications develop. Pulsed Gradient Spin

Echo techniques allow the measurement not only of the bulk

diffusion tensor, but of the structure factor of the

sample.1 Editing techniques use pulsed field gradients to

simplify the complex spectra of biomolecules.2 Local

gradient coils allow functional imaging in the human head.3

NMR microscopy can require field gradients much larger and

switched more rapidly than conventional imaging

experiments.4 Localized spectroscopy allows chemical shift

information to be collected from specific voxels in a living

animal.5 This paper will address some approaches for

producing and evaluating pulsed field gradients.

A technique was developed to measure the eddy current

field that persists after a field gradient is switched off

and, based on the measurement, a filter to correct for the

eddy current field was designed. The technique, which

employs a series of experiments based on the stimulated

echo, was then used to evaluate the performance of the



1D. G. Cory and A. N. Garroway, Magn. Reson. Med. 14, 435, 1990.
2D. Brihwiler and G. Wagner, J. Magn. Reson. 69, 546, 1986.
3K. K. Kwong et al., Proc. Natl. Acad. Sci. 89, 5675, 1992.
4Z. H. Cho et al., Med. Phys. 15, 815, 1988.
5H. R. Brooker et al., Macn. Reson. Med. 5, 417, 1987.









filter. If an eddy current field persists during the period

when the NMR signal is detected, distortions in the spectrum

or image will result. The distortions are particularly

severe when chemical shift information is obtained in the

same experiment as spatial localization by encoding spatial

information in the phase of the NMR signal. It is important

to be able to measure the residual gradient field, which is

usually due to eddy currents in the metal structures of the

magnet, so that it can be corrected by changing the drive to

the gradient amplifier, or by whatever other technique is

available, and to evaluate the remaining uncorrected field

to estimate the distortion that will result in a desired

experiment.

One way to avoid eddy currents for experiments such as

spatially selective spectroscopy is to employ actively

shielded gradient coils. Another, much simpler, approach is

to reduce the size of the gradient coil so that it is widely

separated from the eddy-current-producing structures in the

magnet. This approach is only possible when the clear bore

of the magnet is much larger than the volume of interest,

which is often the case. To make possible experiments, such

as spatially selective spectroscopy, that require rapidly

switched high intensity field gradients, I developed pulsed

field gradient systems based on reduced volume gradient

coils for a 2 T, 31 cm bore magnet used for small animal

studies. This magnet was replaced with a 4.7 T, 33 cm bore

magnet, and the gradient systems were adapted accordingly.









These pulsed field gradient systems offer much better

performance than the large and unshielded gradient system

supplied with the magnet, given their limitation on sample

size. I also developed a pulsed field gradient coil for a 7

T, 51 mm bore magnet used for NMR microscopy and

spectroscopy.

Another experiment which requires gradient coils to

perform exceptionally well is functional imaging of the

human brain. The head is much smaller than the whole-body

magnets in general use. A smaller coil can allow faster

switching to higher gradient fields, as well as reduce eddy

current fields. In order to get a gradient coil that is

matched to the size of the head, some provision must be made

to allow for the shoulders. Conventional designs, even

existing designs with a large linear volume, have current

return paths arrayed on both sides of the linear volume. A

coil matched to the size of the head would not fit over the

shoulders. A coil that trades radial linear region for

increased axial linear region is more appropriate. A design

utilizing concentric return paths was developed that

significantly improved the axial region of linearity. A

prototype was constructed and tested.

In order to perform NMR microscopy and pulsed field

gradient experiments, we adapted an NMR spectrometer and

probe for a 7 T, 51 mm bore magnet. The instrument included

a simple amplitude modulator to carry out slice selection.

A probe that allowed sample loading from above was






4


constructed. Artifacts were eliminated from the images. A

software interface that allows the user to set up an

experiment by entering values in a spreadsheet was

developed. Useful contrast was obtained on fixed biological

samples. Preliminary imaging experiments on both biological

and nonbiological systems were carried out.














MEASUREMENT OF EDDY CURRENT FIELDS


Introduction

It is well known that, when a current pulse is passed

through a field gradient coil in a superconducting magnet,

eddy currents are produced in the conducting structures of

the magnet. Experiments such as diffusion-weighted imaging6

and multiple-quantum spectroscopy7 require that the eddy

current field be a much smaller fraction of the applied

field than do conventional spin-echo magnetic-resonance

imaging experiments. Strategies to reduce the eddy current

field consequently become increasingly important. The two

effective strategies are signal processing of the gradient

demand, known as preemphasis, and self-shielding of gradient

coils, which greatly reduces the interaction of the coil

with the metal structures of the magnet. Often, the two

techniques are used together. When the sample or subject is

substantially smaller than the magnet, another approach is

to minimize the size of the gradient coil. In order to

evaluate and improve the effectiveness of these three

strategies, it is desirable to have a technique to measure

eddy current fields. To implement the preemphasis, it is

necessary to measure the eddy current field in order to


6D. G. Cory and A. N. Garroway, Magn. Reson. Med. 14, 435, 1990.
7C. Boesch et al., Magn. Reson. Med. 20, 268, 1991.








cancel it. An eddy current measurement technique is also

useful in order to evaluate the possibility of performing a

given experiment with available hardware. In this chapter,

a technique for measuring and analyzing the time behavior of

eddy current fields is developed and experimental results

are presented. Some general physical considerations of eddy

currents are discussed, and existing techniques for eddy

current field measurement are reviewed.

An introduction to the Bloch equations will be

preliminary to a discussion of the effect of the eddy

current field on the nuclear magnetization. The Bloch

equations provide a phenomenological description of some

aspects of the behavior of spins in a magnetic field. Let M

be the bulk nuclear magnetization, y the gyromagnetic ratio,

By the polarizing magnetic field, and B1 the amplitude of

the radio frequency excitation field which has rotational

frequency o. T1 and T2 are the time constants associated

with longitudinal and transverse relaxation, respectively.

Mx = Y(BoMy + BlMz sin (ot) Mx/T2 [1]

My = Y(BiMZ cos Ct BoMx) My/T2 [2]

Mz = -y(BlMx sin Cot + BiMy cos Cot) (Mz Mo)/IT [3]

Instead of T2, the symbol T2* is used to denote the time

constant of apparent transverse relaxation when

inhomogeneity in BO is present. Neglecting the effects of

T1 and T2 and assuming B1 = 0, the equations can be

simplified.









MX = TMyBo [4]

My = -'YMBO [5]

Mz = 0 [6]

We can introduce a complex transverse magnetization M = Mx +

iMy so that

M = -iyMB0. [7]

Assume that Bo consists of a constant and a component

linearly dependent on position: B0 = BO+gx. BO is

independent of time and space, while g is quasi-static. If

we define m, the magnetization in the rotating frame, by

M = me-iBOt [8]

then

M = -iyBOM + me-iyBot [9]

Substituting back into Equation [7] gives

-iyBOM + ine-iyBot = -iyM(BO + gx). [10]

Simplifying Equation [10] yields

ne- iyBO t = -iyMgx. [11]

Combining Equation [11] with Equation [8] yields

m = -iygxm, [12]

which has the immediate solution

-iyxft gdt'
m(t) = m(to)e t [13]

If the magnetization has been prepared to a non-zero m(to)

by a radio frequency pulse, the evolution described by

Equation [13] is called a free induction decay (FID).









Consider the characteristics of a general eddy current

field. The eddy currents give rise to a magnetic field that

roughly tends to cancel the applied field of the gradient

coil. The spatial dependence of the eddy current field is

not exactly the same as the applied gradient field.8

The time behavior of the eddy current field is a

multiexponential decay, which can be seen by considering the

form of the solution to the differential equation governing

the decay of magnetic induction due to current flow in the

conductor. Maxwell's equations9 in a vacuum in SI units are

V B = 0 [14]

V *E = [15]
aB
V X E + -- = 0 [16]
at

V X B oF00 DE = o0J [17]

where B is the magnetic induction and E is the electric

field, p is the charge density, e0 and go are the

permittivity and permeability of free space, and J is the

current density. We also assume Ohm's law, J = GE, where y

is the conductivity, assumed to be isotropic and

homogeneous. Taking the curl of both sides of Ampere's law,

Equation [17], neglecting the displacement current, and

using the identity



8R. Turner and R.M. Bowley, J. Phys E: Sci. Instrum. 19, 876, 1986.
9J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York,
1975.









V x (V x A) = V(V .A) V2A [18]

gives

-V2B = g0V X J. [19]

Using Ohm's law to eliminate J for E, neglecting the

displacement current, yields

-V2B = 0CFV X E, [20]

so Equation [16] allows this to be expressed as

rB
V2B = 0a [21]
at
The decay of the magnetic induction must be a solution to

this diffusion equation. Separation of variables gives

solutions for the time part having an exponential time

dependence. This makes it possible to correct for the

linear spatial term in the eddy current field with a linear

filter network. Such a network is known as a preemphasis

circuit.

In a superconducting magnet, the conducting structures

involved are often at very low temperatures and hence have

much greater conductivity than might otherwise be expected.

For example, pure aluminum at 10 K has a resistivity of

1.93 x 10-12 J2-m, while at a room temperature of 293 K its

resistivity10 is 2.65 x 10-8 Q-m. The time scale of the

eddy current decay is directly proportional to its

conductivity, as can be inferred from Equation [21], so eddy

currents will persist 13,700 times longer in an aluminum


10D. R. Lide, (Ed.), CRC Handbook of Chemistry and Physics, 72nd
Edition, CRC Press, Boca Raton, 1991.









structure at 10 K than one at 293 K. In a commercial

aluminum alloy the conductivity will vary from that of the

pure metal, especially at low temperature, so the effect may

not be as great. In practice, the principal source of eddy

current fields is generally the innermost low temperature

aluminum cylinder, which is at approximately the boiling

point of liquid nitrogen, 77 K. The resistivity of aluminum

at 80 K is 2.45 x 10-9 Q-m, so the time constant is about

11 times greater than it would be at room temperature.

We consider the desirable characteristics for an eddy

current measurement technique. Our primary goal will be to

measure the eddy current field in order to evaluate the

feasibility of performing a given experiment, not to

compensate for the eddy current field. Therefore dynamic

range is more important than absolute accuracy. It must be

possible to measure eddy currents produced by specific pulse

sequences, probably by appending the eddy current

measurement experiment to the end of the sequence under

evaluation. It is also preferable to have a technique that

is insensitive to inhomogeneity so that no swimming is

necessary. Since the shim coil power supply may respond

dynamically to the gradient pulse and distort the measured

eddy current field, it is useful to be able to turn the shim

supply off. Experiments based on Selective Fourier

Transform11 and other chemical shift imaging techniques rely


11H. R. Brooker et al., Maan. Reson. Med. 5, 417, 1987.









for spatial localization on the integral of the eddy current

field, so it is desirable to have a measurement technique

that is based upon the integral of the eddy current field.

If possible, the technique should have no special hardware

requirements.


Literature Review

Many workers have addressed the problem of eddy current

measurement and compensation in the literature. The two

aspects of the eddy current field to measure are the spatial

and time behaviors. We review publications that include

descriptions of eddy current measurements, although in most

cases the emphasis is placed upon the preemphasis

compensation process and its effectiveness, not the

measurement. The measurement process can be divided into

techniques that detect the derivative of the eddy current

field, those that detect the eddy current field itself, and

those that detect the integral of the eddy current field.

The derivative of the field is sensed by a pickup coil

consisting of turns of wire through which the changing flux

of the eddy current field produces an electromotive force

that is proportional to the rate of change of the field.12

A high impedance preamplifier boosts the signal. An analog

integrator is usually used to convert the measured voltage

into a quantity proportional to the field, although it is

possible to use digital integration. When used in a magnet


12D. J. Jensen et al., Med. Phys. 14, 859, 1987.









at field, the pickup coil is sensitive also to any change in

flux resulting from mechanical motion, which can contaminate

the measurement. Since the field of the main magnet is

generally about four orders of magnitude larger than the

eddy current field and the time scale of mechanical modes is

smaller than that of the longer time-constant eddy currents,

mechanical stability of the coil is crucial. Drift in the

analog electronics is another potential difficulty with the

pickup coil technique. Even with digital integration, the

preamplifier can experience thermal drift on time scales not

too different from the eddy current field. In spite of

these difficulties, pickup coils are simple to use and can

be used effectively to adjust preemphasis compensation.

They are used routinely to correct for eddy currents in

commercial, clinical MRI installations.13

A different approach to measuring the eddy current

field is through its effect on the NMR resonance. One

advantage here is that a pickup coil and its associated

hardware are not needed. These proportional techniques

measure a frequency shift in the NMR resonance that is

directly related to the eddy current field.14 From Equation

[13], the phase of freely-precessing magnetization in the

rotating frame at time t with respect to to can be written

as




13Personal communication, Dye Jensen.
14Ch. Boesch et al., Magn. Reson. Med. 20, 268, 1991.






13

= yxJ gdt'. [22]

The instantaneous frequency o(t), which can be defined as

the rate of change of the phase of 0 by (0(t) = d)/dt is

related to the eddy current field through the Larmor

equation (0 = yB. Magnetic field homogeneity is important

when using this approach, so that the FID will persist long

enough to obtain a meaningful measurement.

In another approach based on the NMR experiment, the

phase of the magnetization 0 is measured at a single point

in time. The phase at that point reflects the integral of

the eddy current field over certain intervals in the

experiment. Since only one point is sampled in each

experiment, many more experiments are required to map the

decay of the eddy current field than with the proportional

techniques. However, T2* and off-resonance effects do not

affect the usefulness of the technique. The experiment

proposed later is a single-point acquisition technique.

All the techniques surveyed were implemented for

unshielded gradient units, although preemphasis is typically

used on systems with shielded gradient sets as well.15

Boesch, Gruetter and Martin of the University Children's

Hospital in Zurich16,17 measure and correct eddy currents on

a 2.35 T, 40 cm Bruker magnet. The unshielded gradient set

has an inner diameter of 35 cm and a maximum gradient of 1


15R. Turner, Magn. Reson. Imaq. 11, 903, 1993.
16Ch. Boesch et al., Magn. Reson. Med. 20, 268, 1991.
17Ch. Boesch et al., SMRM 1989, 965.









G/cm. They use two NMR techniques to measure the eddy

current field. They interactively correct, using a 12 cm

diameter glass sphere filled with distilled water, and they

use no spatial discrimination in order to get all spatial

components. The experiment consists of a 2.5 s gradient

pulse of 0.6 G/cm followed by a train of 8 FIDs. There is a

20 ms delay between the time the gradient is switched off

and the first radiofrequency (RF) pulse. The RF pulses have

a 20 flip angle in order to reduce echo signals. The total

eight FID acquisition time is 200 ms. They solve the Bloch

equation for a sample with a single resonance frequency and

decay constant and extract

yABz(t) = (MydMx / dt MxdMy / dt) / (M2 + My) [23]

as an estimate of time-dependent Bo shift. They claim this

gives enough information for interactive preemphasis

adjustment. The one measurement they publish is of an

already corrected system and shows 7ABz(t) decaying from 2

to 0 ppm as time t increases from 20 to 200 ms. Glitches

are apparent at the ends of the FIDs.

To map the spatial variations, they place a stimulated

echo (STE) imaging experiment following the gradient pattern

of the experiment they want to analyze. The STE sequence is

applied with and without the preceding gradient pattern.

The difference in phase is considered to be due to the time

integral of the eddy current field in the interval between

the first two pulses of the STE sequence. A series of









slices tilted by multiples of 22.50 is obtained from the

same 12 cm diameter phantom. The images were phase

corrected. The phase of points along the z axis and on

circles around the z axis was measured and used as data for

a polynomial regression analysis to determine the

coefficients of the various spatial harmonics. A table of

the harmonic components following a 2.5 second x gradient

pulse of 0.3 G/cm is presented. The delay between the end

of the gradient pulse and the first RF pulse in the three

pulse STE experiment is 20 ms, and the delay between the

first and second RF pulse is 15 ms. The experiment was

conducted following adjustment of the preemphasis unit. In

decreasing order of magnitude, x, z, y, z2, xz2, xz, and x2-

y2 terms were present. The value of the B0 term was not

reported. Note that after x, the dominant terms should be

eliminated by the symmetry of the coil/cylinder system.

Only the x and xz terms would appear in an ideal system.

The presence of terms having even-order in x can be due to

two reasons. First, the terms may really exist due to

asymmetries in the magnet and gradient coil, crosstalk

between amplifiers, etc. Second, the spherical harmonic

analysis is highly sensitive to the point chosen to be the

origin, and the most favorable origin may not have been

employed.

A series of the phase-modulated images is presented as

well, with delays of 5, 20, 50, and 100 ms between the x









gradient and the STE imaging sequence. The images are all

from an already compensated system.

Van Vaals and Bergman of Philips Research Laboratories

in Eindhoven, the Netherlands,18,19 have a 6.3 T, 20 cm

horizontal bore Oxford magnet with 2 G/cm non shielded

gradients leaving a 13.5 cm clear bore. To measure the

eddy currents, they use a 4 cm diameter spherical phantom.

After swimming, they perform a simple "long gradient pulse,

delay 8, RF pulse, acquire" sequence. The gradient is

switched on for typically 3 s, but at least 5 times the

largest eddy current time constant. For various values of

8, the magnet is re-shimmed to maximize the signal during

the first 10 ms of the FID. The difference in shim values

with and without the gradient pulse is interpreted to be a

spherical harmonic expansion of the eddy current field.

Exact values of 8 are not listed, nor are tables of shim

values. Instead, the amplitudes and time constants of the

eddy current fields, as derived by a Laplace transform

technique, are given. Only the B0 and linear terms are

given; presumably only these terms were shimmed.

Jehenson, Westphal and Schuff of the Service

Hospitalier Frederic Joliot, Orsay, France, and Bruker,20

corrected eddy currents on a 3 T, 60 cm Bruker magnet. The

0.5 G/cm unshielded gradient coils had a clear bore of 50



18J. J. van Vaals and A. H. Bergman, J. Magn. Reson. 90, 52, 1990.
19J. J. van Vaals et al., SMRM 1989, 183.
20P. Jehenson et al., J. Macrn. Reson. 90, 264, 1990.









cm. They use the same type of multiple FID sequence as

Boesch, Gruetter and Martin, with an exponentially

increasing sampling interval and 30 sampling points. The

gradient prepulse is 10 s in length. The first FID is

sampled at 1.5 ms after switching off the gradient, and

sampling continues for 4 s using multiple FIDs. They plot

the measured field vs. the time with and without

compensation. They use a 1 mm by 3 mm water-filled

capillary positioned at +/- 5 cm to discriminate Bo and

linear terms. They do not consider crosstalk or higher-

order terms. They use the same Laplace transform technique

as van Vaals and Bergman, but they apply it iteratively to

get better correction.

Heinz Egloff at SISCO (Spectroscopy and Imaging

Systems, Sunnyvale, CA)21 used a pickup coil to measure eddy

current fields. To correct the B0 component of the eddy

current fields, he moved the gradient coils until the field

shift was eliminated.

Riddle, Wilcott, Gibbs and Price22 considered the

performance of a Siemens 1.5 T Magnetom. They measured the

instantaneous frequency do/dt of a 100 ml round flask

(presumably filled with water) following a 256 ms, 0.8 G/cm

gradient pulse. They present plots for imaging and

spectroscopy shims as well as for the gradient pulse. They

endorse do/dt as an indication of shim. It would seem to


21H. Egloff, SMRM 1989, 969.
22W. R. Riddle et al., SMRM 1991, 453.









work only for single-line samples, however. Following the

gradient pulse, the plot of d'/dt contains peaks that are

not explained. They may be an indication of the true do/dt,

or they may be artifacts from beginnings and ends of FIDs.

The sensitivity of the technique as presented here seems to

be about 1 Hz.

Hughes, Liu and Allen23 of the Departments of Physics

and Applied Sciences in Medicine at the University of

Alberta measured the eddy current fields of their 2.35 T, 40

cm bore Bruker magnet. After 57 delays ranging between 500

is and 2.5 s following a 0.2 G/cm gradient pulse the FID was

measured and the offset frequency of the line determined.

They placed a 13 mm diameter spherical water sample at +/-

1, 2, 4 cm along the axes of the radial gradients under

test. A four-exponential fit was applied to all six

locations simultaneously. The shortest time constant was

associated with the amplifier rise time. An interesting

plot shows that the field associated with each time constant

is essentially linear. The Bo fields associated with the

various time constants are different, however, suggesting a

unique isocenter for each time constant.

Zur, Stokar, and Morad24 of Elscint in Israel place a

doped water sample at +/- 5 cm from the center in the

direction of the gradient of the field. A train of 256 FIDs

is acquired after switching off the gradient. Each FID is


23D. G. Hughes et al., SMRM 1992, 362.
24Y. Zur et al., SMRM 1992, 363.









Fourier transformed, bandpass filtered, then inverse

transformed. The instantaneous magnetic field is obtained

from d)/dt. The digital filtering points to a problem with

phase measurements. The low-pass filters required to

eliminate Nyquist aliasing and to improve the signal-to-

noise ratio (SNR) distort the phase of the received signal.

Digital filtering enables one to recover the SNR ratio of a

small bandwidth without significant phase distortion.

Wysong and Lowe25 at Carnegie Mellon and the University

of Pittsburgh measured eddy current fields on a Magnex 2.35

T 31 cm magnet with unshielded gradient coils. A 1 cm

diameter sphere containing water doped to T-T2-~1 ms is

used. A 0.9 G/cm gradient is applied for 1.0 s, then ramped

down in 128 ps. A train of pulses of flip angle n/2 set 1

ms apart is applied for 1 second. One point is sampled for

each FID. With the system adjusted so the FID is in-phase

in the absence of a gradient field, the out-of-phase

component is proportional to sin(yABte-t/2) = yABte-t/2 for

small values of time and gradients.

Keen, Novak, Judson, Ellis, Vennart and Summers26 of

the Department of Physics, University of Exeter, propose

using a phantom slightly smaller than the imaging volume.

Having switched off the gradient, they delay a variable

time, then pulse and acquire the FID. The Fourier transform



25R. E. Wysong and I. J. Lowe, SMRM 1991, 712.
26M. Keen et al., SMRM 1992, 4029.









of the FID represents a projection of the phantom in the

quasi-steady eddy current field. Measuring the distance

between the peaks that appear as edge artifacts gives the

eddy current field.

Teodorescu, Badea, Herrick, and Huson27 at the Texas

Accelerator Center and Baylor College of Medicine measured

eddy current fields in their 4 T, 30 cm superferric self-

shielded magnet. The magnet was operated at 2.19 T. They

follow Riddle et al.28 in their measurement. A small

phantom is placed at various off-center locations. They use

a 0.8 G/cm gradient pulse of 15 ms and a 750 gs rise/fall

time. This is followed by an FID (or a series of them) that

is acquired for 20 ms. They compare this to the result

obtained from a pickup coil.

The eddy current field was measured with a sense coil

and analog integrator by Morich, Lampman, Dannels, and

Goldie.29 They used a Laplace transform approach to derive

correct parameter values for an analog inverse filter to

compensate for the eddy currents. The analog inverse filter

was of conventional design,30 placed at the input of the

gradient power supply. The theory was tested on an Oxford

Magnet Technology whole body superconducting magnet.

The approach is based on the ease with which a linear

system can be analyzed in the reciprocal space s defined by


27M. R. Teodorescu et al., SMRM 1992, 364.
28W. R. Riddle et al., SMRM 1991, 453.
29M. A. Morich et al., IEEE Trans. Med. Imac. 7, 247, 1988.
30D. J. Jensen et al., Med. Phys. 14, 859, 1987.









the Laplace transform. We can understand the calculation as

follows. Assume the gradient field for t>0 in response to a

unit step function is

N
g(t) = 1- ae-tw i i [24]
i=1

The amplitudes ai and time constants Ti can be determined

through a best-fit to experimental data. To determine the

inverse filter, the first step is to deconvolve the step

function to find the impulse response h(t), which can more

conveniently be accomplished by a multiplication in the

complex frequency space, s. The equivalent function G(s) is

obtained by a Laplace transform

1 N a
G(s) = _- N- ai [25]
S is + Wi

Then the impulse response in the s domain, H(s), is found

through the relation

G(s) = H(s)/s, [26]

so that
N
H(s) = sG(s) = 1 N ais [27]
s + wi
i=1 1
is the impulse response. The inverse filter's impulse

response is just the reciprocal of the impulse response of

the eddy currents,


YH(s) = N [28]
s + Wi









The step response of the inverse filter, F(s), is the

convolution of a step function and the impulse response:

1 1
F(s) =- [29]
sH(s) N ais2
S -
S + wi
i=1 i

The amplitudes bi and time constants vi of the inverse

filter can be read directly from the inverse Laplace

transform, f(t), of F(s):

N
f(t) = 1 + bie-t/vi [30]
i=l

The inverse Laplace transform was performed by matrix

inversion for a four-time-constant case using Gaussian

elimination.

Now the appropriateness of these techniques to the

project of following the time evolution of the eddy current

field can be considered. Two of the techniques, those of

Egloff and Morich, involve the use of a pickup coil,

preamplifier, and integrator. We choose to confine

ourselves to NMR techniques. The procedures of Boesch, van

Vaals, Jehenson, Riddle, Keen, Hughes and Teodorescu require

swimming to correct for the inhomogeneity of Bo. The fact

that T2* must be reasonably long also limits the region

where eddy current fields can be measured to well inside the

active imaging volume. Wysong and Zur propose similar NMR

techniques that do not require swimming. In general,

however, it is samples with long relaxation times that are

most sensitive to small eddy current fields, and the use of









a sample with especially short (T~-T2-1 ms) relaxation times

is not an obvious way to detect low-level fields. The T2 of

the sample limits the duration of the interval in which

phase can be sampled. Another drawback is that the trains

of 7/2 pulses will produce stimulated echoes, even if T1 is

on the order of the interpulse separation. However, this

may be the most promising of the techniques surveyed.


Spin-Echo Techniques

Distortions in the phase of spectra spatially localized

with a two-pulse Selective Fourier Transform technique31

were observed by Mareci.32 He observed that the distortions

were reduced by lengthening the echo time, consistent with

the known behavior of field distortions due to eddy currents

induced in the metal structures of the magnet by the pulsed

gradient fields used for spatial localization. We consider

how a series of spin echo experiments identical except for

n/2y it.

RF nA


g


I I I
0 TE/2 TE

Figure 1. Two-pulse experiment with pulsed field gradient.
The long trailing edge of the gradient pulse indicates
distortion due to the eddy current field.


31H. R. Brooker et al., Magn. Reson. Med. 5, 417, 1987.
32T. H. Mareci, Personal communication.









increasing echo time (TE) gives an indication of the eddy

current field distortion as a function of time. Consider

the evolution of the rotating-frame magnetization m in the

presence of the gradient field g illustrated in the pulse

sequence in Figure 1. For O
Equation [13] so

-iyx f gdt'
m(t) = m(O)e 0 < t < TE / 2. [31]

We can also apply the result directly to describe the

magnetization's evolution following the n pulse. Let TE/2+

be the time just after the n pulse. Then


m(t) = m(TE / 2+)e TE / 2 5 t. [32]

The t pulse along x inverts the sign of the imaginary part

of m(t), equivalent to taking the complex conjugate:

iyx JTE/2 gdt'
m(TE / 2+) = m*(0)e 0 [33]

Putting it together gives

iyxITE/2 gdt' -iE/2 gdt
m(t) = m (0)e e TE / 2 < t [34]
[ (TE/2 : ,
iyx gdt- gdt'
m(t) = m*(O)e J JTE/2 J TE / 2 < t. [35]

Measurement of the phase exactly at the center of the Hahn

echo should remove off-resonance effects, whether due to

chemical shift or field inhomogeneity. Now it remains to be

shown that measurements of the phase at a series of echo

times can be used to find g(t). If 00 is the phase without

a gradient pulse applied, then








(TE) 40 = yx E/2 gdt' J/ gdt] = yx2fTE/2 gdt' JTEgdt].

[36]

We define a function G(t) by

G(t) = yxJ gdt' [37]

which simplifies the expression above for ):

O(TE) 40 = yx[2G(TE / 2) G(TE)] [38]

By measuring 00 and measuring ) at a number of echo times,

we hope to be able to extrapolate the function G(TE), whose

rate of change gives the eddy current field. By performing

a series of experiments in which the values of TE are

related by successive powers of two (TEi1j = 2TEi), we can

obtain a series of coupled equations. Using the shorthand

0 (TEi) = 0j,

Oi+1 = y [2G +i] i = 1, 2, [39]

Inverting for G, yields

GI = [(4i+l 0o)/yx + Gi+1]/2 i = 1, 2, ... [40]

For large enough i, Gi = Gi+I, and the equation has an

immediate solution. The remaining Gi can be determined

recursively. The rate of change of G(TE) is the eddy

current field.

Experiments and subsequent data analysis have pointed

to several drawbacks in this approach. The first is that

the echo time TE limits the maximum length of the gradient

pulse. A gradient pulse long in comparison to the eddy

current decay time approximates a step function, which









simplifies the analysis of the eddy current response.33

However, lengthening the TE reduces the time resolution of

the experiment. Placing the gradient pulse before the

excitation pulse as in Figure 2 eliminates the problem and

decouples the length of the gradient pulse from the echo

time.

n/2y nx

RFnA






Id2 0 d3 TE/2 TE/2

Figure 2. Gradtest v.1.2 is a spin-echo experiment for
measuring the eddy current field following a pulsed field
gradient.


The above analysis assumes a point sample. Any real

sample has finite extent and will experience some dephasing,

and associated signal loss, as its phase evolves in the

gradient field. By not subjecting the transverse

magnetization to the gradient pulse but only to the eddy

current field, the dephasing effect is reduced. Another

drawback proved to be that the signal decayed due to T2

relaxation before Gi stabilized. With the gradient pulse

before the excitation, the condition Gi = Gi, could be met

for small values of TE. However, for large TE we could


33M. A. Morich et al., IEEE Trans. Med. Imaq. 7, 247, 1988.









assume that g = 0 while for small TE, g # 0. To solve for

the Gi it is necessary to know one of them in advance, so to

determine Gi for large TE, another experiment was performed.

TE was held fixed at a large value and d3, the interval

between the end of the gradient pulse and the RF excitation

pulse, was varied in steps of TE/2. A system of

simultaneous equations describes the phase obtained by

varying d3 in steps of TE/2, starting with d3 = 0:

O(TE + d3) 00 = yx[2G(TE / 2 + d3) G(TE + d3) G(d3)]

[41]

The problem of signal decay due to T2 is thus circumvented.

This technique could be used by itself or, as we used it,

only to obtain a starting point for varying TE.

A remaining difficulty is the ambiguity of phase

measurement. Phase can be directly measured only modulo

3600, but the accumulated phase in our experiment may be

much greater. One way around this difficulty is to reduce

the applied gradient so that we can be sure that our sample

rate is above the Nyquist limit, so that 4j+1 i < 1800

To get an upper bound that guarantees no phase ambiguity,

assume that the eddy current field has the same amplitude as

the applied field before the n pulse and zero amplitude

following the 7 pulse. Protons process at 4258 Hz/G. To get

a measurement for TE/2 = 512 ms without phase ambiguity

would, for a sample 1 cm from the center, require a gradient

pulse no greater than 0.000229 G/cm. Such a small gradient









pulse would result in no detectable phase accumulation in

practical cases. Experimental experience showed that it was

not simple to choose in advance a gradient amplitude that

would result in measurable phase accumulation, but no phase

ambiguity, at all echo times. Instead, we repeated the

experiment for a series of increasing gradient amplitudes.

For phase changes of less than 3600, the phase doubles as

the gradient doubles. We could keep track of phase

accumulations greater than 3600, thereby decreasing the

minimum detectable eddy current field.


Stimulated Echo Techniques

The stimulated echo (STE) has advantages over the spin

echo as the basis of an eddy current field measurement

experiment. Consider the stimulated echo sequence Gradtste

in Figure 3. The "e" at the end of the pulse sequence name

indicates that this is a stimulated echo experiment. A third

pulse is required to excite a stimulated echo. The

magnetization of interest is flipped into the transverse

plane by the first RF pulse, where it accumulates phase


RF n n





I I I I I
t T t
grad decay 1

Figure 3. Diagram for Gradtste, a three pulse stimulated
echo experiment for measurement of the eddy current field.









shift due to static field inhomogeneity and eddy current

fields. Then, stored by the second RF pulse along the z

axis, the magnetization accumulates no more phase until the

final RF pulse tips it back into the transverse plane. The

phase accumulation due to static inhomogeneity now unwraps,

resulting in the stimulated echo. If tj is long enough,

there is essentially zero eddy current field in the second

T, so the phase accumulated due to eddy current fields in

the first T is preserved.

It is possible to follow the eddy current decay by

incrementing either tdecay or T between experiments. If T is

incremented, the procedure for determining the eddy current

field is similar to that for spin echo experiments. The

phase shift for two experiments with different T is

subtracted to get the integral of the eddy current field in

the time between the earlier and later T. A more direct

approach is to increment tdecay between experiments, keeping

T small. Using this approach, each experiment yields the

integral of the eddy current field over a short interval T.

Dividing by T yields the average eddy current field in the

interval.

Two advantages of the STE are immediately evident. A

single STE experiment can be directly related to phase

accumulation in a single interval, eliminating the need for

the recursive data analysis or simultaneous equations

associated with the spin echo technique. This would also

seem to make the choice of gradient pulse amplitude more









straightforward. Since tI is limited by T1, which is

generally longer than T2, it is possible to sample with

smaller residual gradient field than in the spin echo

experiment.

The eddy current field is subject to a multiexponential

decay. The integral of a multiexponential decay is another

multiexponential decay. We can expect these functions to be

reasonably smooth. That is, if we notice that the phase is

not changing much between delay increments, we could either

increase the delay increment or increase the amplitude of

the gradient pulse. This is a form of adaptive sampling,

since the sampling strategy for the gradient field depends

upon its behavior. The sampling technique should be capable

of following the residual field decay when preemphasis is

used, and in this situation the field will not in general

decay monotonically, since some of the decay components may

be overcompensated. Therefore the adaptive sampling must

also be able to decrease sensitivity when needed.

Since the eddy current field generally changes most

rapidly at short times, varying T to keep the measured phase

shift approximately constant for each value of tdecay yields

less densely spaced measurements when the field is changing

slowly. We have implemented such an adaptive sampling

technique by writing a recursive macro Adgrad in the Varian

MAGICAL language to perform a series of measurements in

which T is varied to "lock" the phase shift to 45. The

macro functions as a command to the Varian program "VNMR"









through which the spectrometer is controlled. Adgrad allows

the automatic measurement of the eddy current field over a

large dynamic range. Forty-five degrees is large enough to

measure with enough precision and yet small enough to

minimize the possibility of aliasing. The values of phase


'user enters
" Adgrad ( t max) "


Figure 4. Flow chart of the macro Adgrad, which executes
adaptive sampling of the eddy current field. The dotted
portion is not part of the macro.









shift A0 and T are easily reduced to a plot of the eddy

current field vs. time. A flow chart of Adgrad is found in

Figure 4. It is most easily explained in the context of the

whole experimental procedure. The user notes the phase of

the STE for an experiment with gph, the value of the

gradient pulse, set to zero. He then selects a combination

of T, tdecay, and gph that results in phase accumulation of

about 450 and acquires an FID. He also removes the file

"phase.out" if it remains from a previous session. Then he

executes the macro Adgrad(tmax, o0), where tmax is the value

of tdecay at which the macro will stop and 00 is the phase

with gph = 0.

Adgrad first calls the macro Calcphase to compute the

phase A0 at the center of the acquisition window (that is

also the center of the FID) for the data already in memory.

Adgrad then stores the values of A0 and T as the first line

in the output file "phase.out." Next, Adgrad tests to find

if tdecay > tmax. If so, it ends the experiment. This

should not occur on the first pass through the test. In the

following two steps, Adgrad sets up the timing for the next

experiment. The new tdecay is set to be greater than the

old by T, to provide for a contiguous series of intervals t.

The new T is set so that if the eddy current field remains

constant, the next measurement will yield a phase A0 of 45.

Now the measurement is started. Following the measurement,

the macro calls itself and the process repeats. When the









tdecay > tmax test is passed, Adgrad returns control to the
operator.

Two other adaptive sampling macros have been developed

for eddy current testing. Adgrad2 changes both T and gph to

lock the phase to 45. The resulting series of experiments

are more closely spaced in time than Adgrad. Using Adgrad2,

it is possible to follow the eddy current field over a wider

range of values than with Adgrad. However, linearity error

in the digital-to-analog converter or nonlinear amplifier

response will be reflected in error in the eddy current

field. Adgradl80 locks the phase to 1800. It can only work

when the phase accumulation is monotonically decreasing yet

never changes sign, which is true for the uncompensated

gradients. Otherwise, Adgradl80 may lose its lock. If the

error in the measured angle is constant, the accuracy of the

technique, when applicable, should be about 5 times better

than for Adgrad or Adgrad2.

In the preliminary data analysis, we assumed that the

eddy current field was essentially constant over the

sampling interval T, so that g = AO/yxT. An Excel

spreadsheet was used to reduce and analyze the data and plot

the results. An example is shown in Figure 5. It is a plot

of the average field in each of the measurement intervals.

Since the field is dropping exponentially, not linearly,

fits to the mean value will have systematic errors. A

better way is to assume a multiexponential decay of the eddy

current field, and then calculate what phase will be








measured in the STE experiment. If we define OI(t) as the

total phase shift from t = 0 to tn for gph = 1, then
n
(D(t) = [42]
i=l
This phase shift is just, for a single experiment,

*((t) = yx g(t'dt' [43]

Now we assume that the eddy current field can be described

by a three-time-constant decay,

g(t) = Ae- tta + Be-t/tb + Ce- c. [44]

Integration gives a function to which the measured phase can

be fit:

D (t) = yx[taA(l e-tta) + tbB(l e-t/tb) + tc( etc)].

[45]

Results

Eddy current measurements were made on several gradient

coils of practical interest. Tests of the Oxford gradient

coils in the 2 and 4.7 T magnets were conducted. For the

4.7 T magnet, the eddy current measurements were used to

adjust the preemphasis network. Measurements of the eddy

current fields associated with home-built gradient coils

were also made. The detailed design and construction of the

coils, on 9 and 15 cm former, is described in the following

chapter.

Initial measurements were made using the spin-echo

technique of Gradtest vl.2. A 5 mm NMR tube with about 5 mm










of H20 trapped by a vortex plug was used as a sample and

placed 1.7 cm from the center of a 2 T, 31 cm horizontal-

bore magnet (as measured from an image). The Oxford Z

gradient in the Oxford 2 T magnet was pulsed to a value of

1000 units or 1 G/cm. The manufacturer-installed

preemphasis filter was in place. A d3 array with four

elements was used to establish the phase value for large

echo times via matrix inversion of simultaneous equations.

An echo time array resulting in a series of coupled

equations was used to work back to 1 ms. The resulting plot

is shown in Figure 5. The bumpiness of the plot may be due

to the preemphasis. Data points are plotted in the center

of the interval for which they represent the average

gradient.



0.045

0.04

0.035

0.03
U
0.025

0.02

m 0.015

0.01

0.005

0 --- I I-Illi'- -
0 100 200 300 400 500 600 700 800 900
t (ms)


Figure 5. Eddy current field as a fraction of applied field
for Oxford gradient coil.









Stimulated echo measurements using the pulse sequence

Gradtste and the macro Adgrad were conducted for the Oxford

gradients as well as for the 9 cm home-built gradient coil

in the 2 T, 31 cm diameter magnet. The eddy currents for

the Oxford gradients were measured with the manufacturer-

installed preemphasis filter in place. The 9 cm coils had

no preemphasis. A 5 mm NMR tube with about 5 mm of H20

trapped by a vortex plug was used as a sample and placed

between 1 and 2 cm from the center of the magnet. The

center of the sample was determined from an image. In all

cases, tgrad = 2 s, dl = 2 s, tj = 0.5 s, and two averages

were acquired. The parameter T was set to 4 ms and tdecay was

1 ms for the initial experiment. The data were analyzed in

Excel spreadsheets. In the plots of field vs. time given in

Figures 6 and 7 for the Oxford and 9 cm coils respectively,

the average of the eddy current field over the sampling

interval is plotted against the middle of each sampling

interval. The eddy current field is represented as a

percentage of the applied gradient. Note that without

preemphasis, the eddy current field due to the 9 cm coil

declines monotonically, while the preemphasis filters

contribute to the measured field of the Oxford coils. For

the 15 cm coil tested in the 2.0 T magnet, eddy current

measurement was used to calculate values for an inverse

filter. The coils and samples were removed between the

experiments before and after preemphasis. The Techron 7540

amplifiers were used to drive the coils in current mode.


























Hm r in

(%) (^)6


_ --,- I


I~ 11 I -


C,, u-


H4 u-
0


S L

I


(%) (4)B


a) 00 r- w


in m (N H-


t


0 H-
I


(%) (4))


S Al 0Co


m in


-mm "m **.


4

*-)



0






,3-
co
0)H












-H




0 -H

0)-


rd
d4
01






o *
O u




44
0) -H






x eo
0
r-
r- O
0 m













00
p H
: 4-4


a) -H
*H (I
(0) 0




rx.4 1


* -


















m
U







.m m
*









U *

W ..


(%) (q)6


r- 0 0 0 0

(%) ( )B


M *
, -- I .


(%) (1)


r-1
-H
-r 0
(s u






S-H
0u
u
( >4



E-H -H


0
0
-A

0



U
-H o
ro
.,-0 )
U2




O rd
a-,-




0(U





Oi -



0(1)



S-H





., --I



>H
-H U

rX :


* N











The same sample and RF coil were used as in the Oxford and 9

cm tests. In all cases, tgrad = 0.5 s, tL = 0.5 s, the

sample position was between 1 and 2 cm from the center, and

the position was measured in an image. The data were


g(t) (%)
5

4

3

2

1


t (s)


0.1 0.2 0.3 0.4


0.5


g(t) (%)
6


4
3
2
1


t (s)


0.2


0.3 0.4 0.5


= t (s)
0.5


0.1 0.2 0.3 0.4


Figure 8. Eddy current field of 15 cm gradient coil set in
2.0 T magnet system before (upper curve) and after
compensation (lower curve). a) X coil; b) Y coil; c) Z coil.


g(t) (%)

6
5
4
3
2
1
0









acquired with Adgrad2, that changes gph as well as T to keep

the phase locked. Eight averages were acquired. The eddy

current field was measured out to 1 s, although the plots in

Figure 8 only show 0.5 s. The data were analyzed both with

the average-field technique used in the Excel (Microsoft,

Inc.) spreadsheet and with a multiexponential curve fit in

Mathematica (Wolfram Research, Inc.). The curve fits seemed

more satisfactory, and are shown in Figure 8. The lower

curves represent the eddy current field after compensation.

The curves plotted are the derivatives of the exponential

curves that were fitted to the raw data. The preemphasis

filter amplitudes and time constants were taken to be those

of the eddy current field itself. This procedure should


0(t) (degrees) Echo Phase Shift



400



300



200



100



t (s)
0.2 0.4 0.6 0.8 1

Figure 9. Fit to raw data of eddy current field of Oxford Z
gradient field for 4.7 T magnet system before compensation.









tend to underestimate the preemphasis required, but since

the unshielded eddy current fields were already less than 5%

of the applied field, the error is not severe.

The 4.7 T magnet that replaced the 2 T 31 cm magnet did

not have manufacturer-installed preemphasis, so the eddy

current measurement techniques were applied to design an

appropriate preemphasis filter. Since the uncompensated

eddy currents were on the order of 50% of the applied field,

the approximation used to compensate the 15 cm coil would

not be effective. An inverse Laplace transform technique

was used to design the filters. The technique was

implemented through the symbolic inverse Laplace transform

capability of Mathematica. An example of a multiexponential

fit to the raw phase accumulation performed with Mathematica

is shown in Figure 9. Eddy current fields before and after

compensation are presented in Figure 10. The upper curves

represent the field before, and the lower curves after,

preemphasis. For the Y coil, the procedure was repeated a

second time to obtain an additional reduction of the eddy

current field. The lowest curve in Figure 10 (b) represents

the eddy current field after the second pass of eddy current

correction.


Conclusion

A technique to measure the eddy current field of a

pulsed field gradient based on the phase of the stimulated-

echo NMR signal has been proposed. Experimental







42

g(t) (%)
50
40
30

20


0 t (s)
a) 0.2 0.4 0.6 0.8 1


g(t) (%)
40
35
30
25
20
15
10
5
0 t (s)
b) 0.2 0.4 0.6 0.8 1

g(t) (%)
40

30

20

10

0 t (s)

C) 0.2 0.4 0.6 0.8 1




Figure 10. Eddy current field of Oxford gradient coil in
4.7 T magnet system before (upper curve) and after
compensation (lower curve). a) X coil; b) Y coil. Lowest
curve was acquired after second-pass preemphasis; c) Z coil.


verification consists of measurements of the eddy current

field before and after preemphasis. The level of the eddy

current field after preemphasis can be interpreted as an

upper limit on the error bar of the measurement. It is only

an upper limit, since other errors also contribute to the









residual eddy current field. Any error in the values of

timing components in the preemphasis filter will add to the

residual eddy current field. Also, any distortion in the

amplifier will reduce the effectiveness of the compensation,

since the filter was designed based on the assumption that

the amplifier is linear. Error in the eddy current

measurement technique itself might be due to other echo

terms than the stimulated echo contributing to the signal.

However, experiments have shown that other echoes are

essentially negligible due to a combination of favorable

timing and phase cycling. In the case of Adgrad2, which

scales gph as well as T, it is clear that some error is due

to inaccuracies in the digital-to-analog converter (DAC)

output level. The applied gradient is then not proportional

to the DAC code, and so there is an error in normalizing to

the applied gradient. Error in the curve fits may be

significant, since in a multiple-exponential fit it is

difficult to get an accurate fit if the time constants are

not widely separated. Note that second-pass adjustment of

the preemphasis was more effective in reducing the residual

eddy current field.

The technique came out of a need to quantify phase

distortions in localized spectroscopy. It is therefore

better-suited to measuring the time integral of the eddy

current field than the field itself, and it is the integral

of the field that gives rise to errors in phase-sensitive

techniques such as SFT. It is often useful to employ the









basic stimulated echo experiment without adaptive sampling

to quantify the integral of the eddy current field over an

interval, and allow one to predict the resulting phase

distortion directly.

The adaptive sampling algorithm is able to follow eddy

current fields that are not simply monotonically decreasing.

I found experimentally that if the angle became much

different than 450, the values for T would bounce around a

lot before stabilizing. This is probably due to the control

being purely proportional. Introducing an integral term

might help.

The relatively large eddy current field produced by the

15 cm Z gradient compared to the X and Y channels is due to

its extended-linearity design, which locates the currents

farther from the region of interest than a Maxwell pair.

The relatively large eddy current field produced by the 9 cm

Y gradient compared to the X and Y channels may be due to a

problem with centering the gradient coils in the bore. The

measured field gradient would then depend strongly on the

position of the sample.34

The contrast in eddy current field between the large

and small coils is clear. There is a factor of about 10 in

eddy current field between the 15 cm coil and the larger

Oxford coil. There is a factor of about 180 in eddy current

field between the 9 cm coil and the Oxford gradient coil set


34D. J. Jensen et al., Med. Phys. 14, 859, 1987.






45


in the X and Z channels. Experimental evidence demonstrates

the advantage in eddy current field obtainable with reduced

size gradient coils.














GRADIENT COIL DESIGN


Introduction and Theory

Although virtually all NMR measurements rely on

auxiliary field coils, there has been comparatively little

published work on the design and analysis of shim and field

gradient coils compared to that for radio frequency coils.

However, high levels of performance have become increasingly

important for these low-frequency room-temperature coils on

several frontiers of the NMR technique. Three of these

areas are gradient coils for NMR microscopy, coils for

spatial localization of spectra, and local gradient coils

for functional imaging of the human brain.

The simple forms of discrete element coil designs have

linear regions that are about 1/3 of the coil radius.35

Therefore the gradient coil must be considerably larger than

the sample. Increasing the linear region would allow

smaller coils to be used, generally improving efficiency and

decreasing eddy current fields. Several approaches are

available to increase the region of linearity. Adding

discrete elements to cancel more high-order terms in the

harmonic expansion has been done successfully by Suits and

Wilken.36 Continuous current density coils have also been


35F. Romeo and D. I. Hoult, Magn. Reson. Med. 1, 44, 1984.
36B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.









designed with linear regions that are a large fraction of

the radius.37 We have tried to take a fresh approach,

combining aspects of both continuous and discrete designs.

For a solenoidal main magnet, available radial gradient coil

designs are longer and less efficient than axial designs, so

we have chosen to concentrate on the radial case.


z

z


e
/ 0 y



X

Figure 11. The coordinate system used in the text.


An appropriate starting point to find a new radial

gradient coil design might be: what current distribution on

the surface of an infinitely long cylinder would produce a

field in which the axial component is linearly proportional

to the radial position, Bzo x? To describe surface

currents and fields, we introduce the three coordinate

systems described by Figure 11. Any point can be described

in any of three orthogonal coordinate systems. In the

Cartesian system a point is described by its location along

the three axes (x, y, z). In the spherical system, it is

described by two angles and the distance from the origin:


37R. Turner, J. Phvs. D: ADpp. Phys. 19, L147, 1986.









(r, 0, )) In the cylindrical system, the point is

described by (p, ), z). It can be easily shown that an
azimuthal component of the surface current, J0, proportional

to cos) and independent of z produces the desired spatial

dependence. Neglecting for the moment the problem of

current continuity, there are two possible approaches to

achieving the cos) angular dependence. First, it can be

approximated by superimposing azimuthal currents with no

axial component. The solutions are exactly the same as for

discrete filamentary currents. The first approximation, the

1200 arc familiar from the so-called Golay double-saddle

design,38,39 is shown in Figure 12(a). This class of

designs has been called the "Golay Cage" because of its

correspondence to the double-saddle design. Higher-order

approximations utilizing superimposed arcs are derived by

Suits and Wilken.40 The other approach is to use our

freedom to choose any axial current to meet J4o cos# by

varying the current direction, for example, J4Oc cos#), Jzo

sin). This approach leads to the Cosine Coil shown in

Figure 12(b). Note that in Figure 12 the return paths are

located away from the active volume of the coil. For a coil

of practical length, the current return paths can

significantly reduce and distort the gradient field.




38F. Romeo and D. I. Hoult, Magn. Reson. Med. 1, 44, 1984.
39M. J. E. Golay, Rev. Sci. Inst. 29, 313, 1958.
40B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.






















x x

(a) (b)


Figure 12. Two radial gradient coils, a) The Golay Cage
Coil; b) Cosine Coil.


Another approach to current return paths is possible if

we relax the requirement that the current is confined to

the surfaces of cylinders. The current return paths can be

located in the same plane as the azimuthal current paths. A

gradient coil can be constructed of a stack of planes

approximating a current sheet, such as shown in Figure 34

(a) on page 111. The planes include radial as well as

azimuthal current elements. The radial currents do

contribute to the axial magnetic field. It happens that the

third-order harmonic terms eliminated by using 1200 arcs are

independently zero for the radial currents connecting the

arcs. These Concentric Return Path (CRP) Coils can have a

linear region that can be increased in length by stacking

more planes together. The overall combined coil structure

can also be very short, since the return paths do not

require extra length. In order to improve the linear region









beyond that produced by a constant current density along z,

we can adjust the relative current or position of each

planar unit.


Literature Review

This literature review will be focused on efforts to

increase the useful volume of a gradient coil, to optimize

its performance, and to understand the eddy current field

associated with a switched gradient it produces. The

specific requirements of coils of interest for functional

imaging of the human head are discussed, along with several

approaches to meeting those requirements.

Gradient coils can be grouped into two broad

categories: those made up of discrete current elements as in

Figure 13, and those approximating a continuous current

density. The former include the original NMR shim coil

designs,41'42 while the latter approach has been used to

make possible actively shielded gradient coils.43

Anderson described a set of electrical current shims

for an NMR system based on an electromagnet.44 The coils

were located in two parallel planes, one against each

poleface, to allow access to the sample. Each coil was

designed to produce principally one term in the spherical

harmonic expansion of the field. The orthogonality of the



41W. A. Anderson, Rev. Sci. Inst. 32, 241, 1961.
42M. J. E. Golay, Rev. Sci. Inst. 29, 313, 1958.
43P. Mansfield and B. Chapman, J. Maan. Reson. 66, 573, 1986.
44W. A. Anderson, Rev. Sci. Inst. 32: 241 1961.









expansion ensured relatively independent adjustment of the

current in the various coils.

Techniques for designing higher-order shim coils for

solenoidal magnets were set forth by Romeo and Hoult.45

Coils are designed by expanding the Biot-Savart integral for

Bz, the axial component of the field, in a spherical

harmonic series about the center of the coil for simple

filamentary building block currents.
1
Bz(r,9,()) = XAl,mPi(cos )ei. [46]
1=0m=-1
The functions Pmf(cos ) are the associated Legendre

functions. As building blocks are added in the form of arcs

on the surface of a cylinder, more terms in a spherical

harmonic expansion of the field can be set to zero. The

designer connects the building blocks in such a way as to

satisfy the requirement of current continuity, which is not

built into the Biot-Savart law. By setting each undesired

term in the harmonic series to zero, a system of equations

results. The solutions are the current, length, and

position parameters of the coil designs. A Maxwell pair, as

shown in Figure 13(b), is composed of a loop placed at 0 =

600 and another having opposite current direction placed at

8 = 120. This separation is required to cancel the (1, m)

= (3, 0) term, while the odd symmetry cancels the (2, m) and

(1, m # 0) terms. The desired (1,0) term remains. The


45F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984.









simplest coil producing a gradient perpendicular to the axis

of the cylinder is the double saddle or "Golay" coil

illustrated in Figure 13(a). The arcs all subtend 1200 and

are placed at the four angles 01 = 68.70, 02 = 21.30, 1800-1,

and 1800-2, where they produce an (1, m) = (1, 0) term but

no (1, m) = (3, 0) term. The relative current directions

are shown in Figure 13(a). A family of solutions exists for

which the sum of the (1, m) = (3, 0) terms produced by the

arcs cancels, but the (1, m) = (3, 0) terms produced by each

arc are not necessarily zero. We designate such coil

designs by the two angles 01 and 02, so that the design

above would be described as 68.70/21.30. Adding additional

current elements to the coils adds degrees of freedom to

the system of simultaneous equations, and makes it possible

to cancel more terms.

Adding another pair of loops adds two more degrees of

freedom (the current and position of the new loops), and

makes it possible to cancel higher-order terms including (5,

0). Note that the equations are not linear, so for a large

number of current elements the procedure becomes unwieldy.

For shim coils, it is less important to improve the linear

region of a first-order or gradient coil than to design

additional coils whose lowest-order terms are of

increasingly high order. The simple saddle coil in Figure

13(a) designed by this technique has a useful volume with a

radius of about 1/3 that of the cylinder.





















x x

(a) (b)


Figure 13. Field gradient coils that use discrete
filamentary current elements. a) Double-saddle 68.70/21.30
coil to produce the radial field gradient x or y; b)
Maxwell pair produces the axial, or z, field gradient.


The approach is best suited to cases where the gradient

coil is much larger than the sample, since the harmonic

series approximation to the field converges more rapidly

near the center of the coil. Although in theory the current

elements are lines, in practice they do have finite

dimensions, especially where large field intensity is

required. Including the wire diameter would greatly

complicate the design procedure. It is natural to use this

technique to design the shim coils mentioned above, where it

is conventional to have separate adjustments for as many as

twelve or more terms in the harmonic series. The coils

designed this way have the advantage of simplicity of

construction.









The building block approach was successfully extended

by Suits and Wilken46 to use discrete wires to produce a

constant field gradient over an extended region. They

evaluated designs for cylinders with the polarizing field

both parallel and perpendicular to the axis. To improve the

useful volume of the radial gradient coil, they superimposed

four saddle coils. The available degrees of freedom then

included the number of turns in each of the four coils, the

angular width of the four arcs, and the axial positions of

the arcs. Systems of nonlinear equations result that were

solved to null desired terms in an expansion of the field in

orthogonal functions. Numerical plots demonstrate that the

useful volume was extended to about eight times that of the

simple saddle coil. In each case, the volume was

nonspherical. The problems of extending this approach

further are that larger systems of nonlinear equations are

increasingly difficult to solve, and that the orthogonal

expansions do not converge rapidly away from the center of

the coil.

Bangert and Mansfield47 designed a gradient coil in

which the wires were included in two intersecting planes.

The wires in each plane formed two trapezoids symmetrically

placed about the z axis. By setting the angle between the

planes to 450, the third-order terms in the magnetic field

are canceled.


46B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.
47V. Bangert and P. Mansfield, J. Phys. E: Sci. Instrum. 15, 235, 1982.









Approaches based on a continuous current density are

most often used for gradient coils, where performance and

linear volume are more important than simplicity. The

impetus for these coils was the echo-planar imaging

technique of Mansfield,48 which requires high intensity

field gradients switched about an order of magnitude faster

than conventional Fourier imaging. Also, shielded coils are

useful in other imaging experiments that require rapidly

switched gradient fields, and in volume localized

spectroscopy. Without shielding to cancel the external

field, the higher frequency and intensity lead to greater

eddy currents in the cryostat and magnet, that in turn

distort the linearity and time response of the field. Using

current on two concentric cylinders, it is possible to

produce a linear field inside the inner cylinder and zero

field outside the outer cylinder. Continuous current

density coils can be designed to have a large linear region,

and, since current flows on the surface of the whole

cylinder, high efficiency.

A Fourier transform technique was applied by Turner to

design gradient coils that approximate a continuous current

distribution. The approach arose from consideration of the

eddy currents induced on cylindrical shields concentric to

gradient coils made up of discrete arcs.49 Expansion of the

Green's function in cylindrical coordinates was a natural


48p. Mansfield and I. L. Pykett, J. Macn. Reson. 29, 355, 1978.
49R. Turner and R. M. Bowley, J. Phys. E: Sci. Instrum. 19, 876, 1986.









approach to calculating the eddy current distribution. It

was then possible to write the field produced by a general

current distribution on the surface of a cylinder as a

Fourier-Bessel series.50 An inverse Fourier transform of

the Fourier-Bessel series allowed the current to be

expressed in terms of the desired field on the surface of an

imaginary cylinder. The field must satisfy Laplace's

equation to allow the existence of the inverse Fourier

transform. So by specifying the inverse Fourier transform

of the desired field, the current distribution required to

generate that field could be calculated. The continuous

distribution of current is approximated by discrete wires.

The wires are placed along the contour lines of integrated

current. Although the principal application of the

technique was shielded gradients, unshielded coils having

extended linearity were also designed. For example, a

radial gradient coil is reported to have a gradient uniform

to within 5% over 80% of the radius and a length of twice

the radius. The overall length of the coil is about 9 times

the radius.

It was pointed out by Engelsberg et al. for the case of

a uniform solenoid that the homogeneity of the coil depends

strongly on the radius of the target cylinder.51 They note

that the field has the target value only on the surface of

the target cylinder. For example, in order to achieve a


50R. Turner, J. Phvs. D: Appl. Phys. 19, L147, 1986.
51M. Engelsberg et al., J. Phys. D. 21, 1062, 1988.









homogeneous field along the axis of the solenoid, the target

cylinder should be as narrow as possible. The effect is

especially pronounced at the ends of the target cylinder.

The importance of functional imaging of the human brain

and its reliance on the echo planar imaging technique puts

special demands on the rise time and field of the gradient

coil. The fact that smaller coils will be more efficient

and less affected by eddy currents has motivated several

workers to design gradient coils that will fit closely over

the head. To use a small gradient coil it is necessary to

have extended linearity in the radial and axial directions.

For a head coil, extended axial linearity is especially

important to allow the diameter of the coil to be smaller

than the width of the shoulders.

Wong applied conjugate gradient descent optimization to

the design of gradient coils with extended linearity.52 He

allows the position of current elements to vary to minimize

an error function. It is possible to define the error

function as desired, so it is simple to optimize over

regions of any shape, or for coil former of any shape. It

is also simple to include parameters such as coil length.

Repeated numerical evaluation of the Biot-Savart law for the

test wire positions would limit the application to coils

with a fairly small number of elements. Wong applied the

technique to the design of a local gradient coil for the


52E. C. Wong et al., Maan. Reson. Med. 21, 39, 1991.









human head.53 Its overall length was 37 cm, diameter 30 cm.

The region of interest is a cylinder 18.75 cm in diameter

and 16.5 cm long, over which the RMS (root mean square)

error in the field was less than 3% for all three axes. The

gradient coil was symmetric to avoid torque. In order to

make still shorter coils, Wong placed the return paths on a

larger cylinder.54 The wires on the inner cylinder were

connected to the return paths on the outer cylinder over

both endcaps. A coil was designed of 30 cm length, 30 cm

inner diameter, and 50 cm outer diameter. The optimization

region was a cylinder 24 cm long and 20 cm in diameter, and

the RMS error over the cylinder was 7.2%. The symmetry of

the coil eliminated the torque that arises in other short

designs. Additional points on a cylinder 70 cm in diameter

were added to the region of interest to force some partial

shielding.

Another approach to the design of gradient coils that

will fit over the head is to design a coil that has its

linear region at one end. Myers and Roemer55 used only half

of a conventional coil to move the linear region to the end.

A target field approach was used by Petropoulos et al. to

design an asymmetric coil with low stored energy.56 The

coil simulated was 60 cm long and 36.4 cm in diameter. The

"center" of the coil was 14.5 cm from one end. The stored


53E. C. Wong et al., SMRM 1992, 105.
54E. C. Wong and J. S. Hyde, SMRM 1992, 583.
55C. C. Myers and P. B. Roemer, SMRM 1991, 711.
56L. S. Petropoulos et al., SMRM 1992, 4032.









energy for a gradient of 4 G/cm was calculated to be 7.93 J.

Since these coils can be made much smaller than the bore of

the magnet, eddy currents are not a serious problem and

neither of these coils is shielded. Unlike symmetric

designs, these coils experience a net torque in the magnetic

field that is potentially dangerous.

Another coil at a larger radius can be used to cancel

the torque experienced by an asymmetric gradient coil.

Petropoulos et al.57 took this approach to design a head

coil with an inner diameter of 36.4 cm, the same as their

single-layer coil described above, and an outer diameter of

48 cm. The length of both inner and outer coils was 60 cm.

The coil was designed to have a useful region that is a

sphere of 25 cm diameter. There is a price to pay in

increased stored energy, which increases over the single

layer coil value of 7.93 J to 19.2 J. Torque-compensating

windings can be added to the same cylinder as the primary

coil, resulting in a long structure one end of which is

placed over the head of the patient. Abduljalil et al.58

developed such a coil set for echo-planar imaging. The

diameter of the two radial coils was 27.2 cm and 31.2 cm.

The center of the linear region was 17 cm from the end. The

overall length was not reported, but based on artwork for

the wire pattern, it seems to be about 116 cm.




57L. S. Petropoulos et al., SMRM 1993 1305.
58A. M. Abduljalil et al., SMRM 1993, 1306.









Turner has suggested that the best approach to a

compact gradient head coil design is that of Wong, in which

the return paths are placed on a larger cylinder.59 He

points to the trapezoidal gradient coil designed by Bangert

and Mansfield,60 and discussed above, as a starting point

for this approach. The concept for such a gradient coil is

described in a patent by Frese, for a cylindrical

geometry.61 It can be thought of as a Bangert and Mansfield

coil in which the inner and outer wires have been stretched

into arcs on concentric cylinders. This is the design

independently developed by Brey and Andrew and dubbed the

Concentric Return Path Coil (CRPC). Frese suggested using a

stack of the planar CRPC units with spacing along the

cylinder's axis varied to improve size of the linear region.

He also suggested that the angle of the arcs could be varied

from plane to plane. No specific information on the spacing

or angle of the arcs is provided.

A survey of the literature suggests that it is

desirable to design a short gradient coil using the basic

concentric return path geometry to be used for the human

head. A direct error-minimization technique is appropriate

for two reasons. First, the Fourier-Bessel transform

technique, although computationally efficient, limits the

shape of the region of optimization to the surface of a



59R. Turner, Maan. Reson. Imaa. 11, 903, 1993.
60V. Bangert and P. Mansfield, J. Phys. E.: Sci. Instrum. 15, 235, 1982.
61G. Frese and E. Stetter, U. S. Patent 5,198,769, 1993.









cylinder, and axial linearity is important for the head

coil. Second, the currents are not confined to the surface

of a cylinder, and the transform technique in its present

form allows only for current on the surface of a cylinder.


Field Linearity

An extended linear region is one of the goals of a

reduced-size gradient coil. In order to evaluate a coil

design in terms of its linear region, it is necessary to

define the boundary of the linear region. An appropriate

definition for the error associated with a field gradient

reflects the purpose of the gradient coil. In an error

minimization technique, the error definition is central to

the coil design. A reasonable parameter to use is the error

in the field, B.E.= Bz() where the desired gradient, G, is
G x

measured at the center of the coil. Another error parameter

is the error in the gradient, G.E., defined by

1 dBz(x)
G.E.= In an NMR image, error due to the
IGI dx

gradient coil simply produces an error in the mapping

between the sample and the image. The absolute mapping

error is simply the error in the field, B.E. In practice,

samples are usually centered in the gradient coil, so we may

want to weight the error toward the center of the coil. We

use an error parameter that corresponds to the mapping shift

relative to the component of the distance to the center in









the direction of the gradient, the relative error

B (x)- G x
R.E., defined by R.E.= Bz)- G
G x


Efficiency

In order to make use of the efficiency the reduced size

of an extended-linearity gradient coil design can provide,

it is necessary to construct the coil in such a way that it

can be driven efficiently by an amplifier. By adjusting the

number of turns, it is possible to trade maximum gradient

for switching time. We will show that to obtain optimal

switching time, the amplifier should be current-controlled

to compensate for the inductance of the coil. To reduce

switching time with such an amplifier, the coil resistance

per turn should be as low as possible, even though the time

constant of the coil will be lengthened. A time-domain

model for the coil and amplifier will be used to explore the

tradeoff between maximum gradient and switching time.




L

c
R,



Figure 14. Time-dependent voltage source v(t) drives
inductive load.


We show that a current-controlled amplifier gives

better switching performance into an inductive load than a









voltage source. The amplifier, modeled by a time-dependent

voltage source, v(t), is connected to a load with

resistance, Rc, and inductance, Lc, as shown in Figure 14.

When a demand is applied to a current-controlled amplifier

for some current, io, it will by definition change its

output voltage, v(t), as much and as rapidly as possible to

change the current through an inductance across the output.

If the maximum output voltage of the amplifier is Vo, and we
define the steady state output voltage v0 = Vo / Rc, where

Vo>v>O, then the amplifier output voltage and current as a

function of time will be

0 t < 0 0 t < 0
v(t) Vo 0 < t < to i(t) = I. 1- e-t/ 0 < t < to
Vo to < t < t Rc
i to < t<


[47]

where to is the time at which the output current reaches the

desired current io, and T = Lc/Rc. It is straightforward to

calculate that

to = TIn Vo [48]

The smaller the ratio of V0 to vo, the greater the switching

time to will become. If the amplifier is a voltage source,

the desired current will never be exactly reached. It is

more desirable to use a current-controlled amplifier for

which o>>v0.









It will be shown that, with a current-controlled

amplifier, additional series resistance per turn always

decreases performance. Therefore, the series resistance

should be reduced as much as possible, for example, by using

a larger cross-sectional area for the winding in a coil of

fixed radius. Consider again the time response of a

current-controlled amplifier from Equation [47]. Any

internal amplifier resistance can be included in Rc to avoid

any loss of generality. Assume there is some finite,

positive Rc that maximizes i(t). Then for that
di
R, = 0. Solving for Rc:
dRc

di V + Vo [491
di =_ i-e- L-+ -e- = 0 [49]
dRc Rc2 c c L

and assuming that Rc and Vo do not vanish,


1 e + R/c L e = 0. [50]


This can be written as

(1 + x)e-x = 1 where t/T = x, [51]

therefore

ex = + x. [52]

There is no positive value of x that satisfies Equation
di
[52]. Hence < 0 for all t>O, Rc>O. A lower Rc is
dRc

always an advantage when using a current-controlled

amplifier, although the time constant T = Lc/Rc of the

gradient system increases as Rc decreases. It is then









appropriate to maximize the cross-sectional area of the

windings subject to considerations of linearity and

available space.

Next we consider how the number of turns of wire in a

coil of fixed cross-sectional area can be varied to achieve

the desired performance. It is important to note that the

time constant of a coil, for a fixed area, can properly be

considered to be independent of the number of turns. This

result follows from consideration of a gradient coil at low

frequencies, as described by the equivalent circuit of a

series resistor Rc and inductor Lc as shown in Figure 15.




LC


Re



Figure 15. Equivalent circuit of a gradient coil in the
low-frequency limit.


Let R1 be the resistance, and let L1 be the inductance of a

single turn coil. It is well known that the inductance of a

coil increases as the square of the number of turns of wire

N.62 The resistance also increases as the square of the

number of turns if the area is held constant, since as the

number of turns increases, the area of each turn diminishes.



62T. N. Trick, Introduction to Circuit Analysis, p. 256, John Wiley and
Sons, New York, 1977.









The total resistance and inductance are then

Rc = R1N2 Lc = L1N2. [53]

The time constant T of the coil is just the ratio

S= Lc/ R [54]

Perhaps surprisingly, T is independent of N. The result

does not apply when additional turns of wire are added to an

existing coil, thus increasing the area. However, since the

area should already be as large as possible to maximize the

performance, it will not be possible to increase N without

decreasing the size of the wire.

To determine how many turns of wire N should be used in

the gradient coil, we consider how rapidly and to what value

the current rises for various N, holding the area constant.

It will be shown that with a current-controlled amplifier,

the coil is optimized to switch to the field at which it

reaches a saturation current, IO, which is the maximum that

the amplifier can supply. Consider an amplifier with

negligible output impedance switching at time t = 0 from

zero current to maximum current, I0, through a gradient

coil, reaching I0 at to. Assume that Rc < VO/IO. The

current as a function of time is:


1-e1 0 t 5 to
i(t) = Rc [55]
TO t > to

We define a current efficiency k for a single turn so that

the gradient field G(t) = kNi(t). Rc varies as N2, and the









magnetic field G varies as N, so

kVo -/
f Wo1 e- 0 < t < to
G(t) = RIN [56]
kNIo t 2 to

A plot of G(t) for various values of N is shown in Figure

16. All three curves have the same time constant, so


G(t)


0.175

0.15

0.125

0.1

0.075

0.05

0.025


0.1 0.2 0.3


Figure 16. Magnetic field produced by a current controlled
linear amplifier coupled to a coil of fixed dimensions.
Each curve represents a different number of turns.


the difference in slope is due to the relative amplitude of

the maximum gradient. The dotted line connects all the

current-limit points. Since at the current-limit point to

the amplifier is both a voltage and a current source, we can

eliminate G(to) in favor of N and to, yielding an optimum

number of turns for a given switching time.








Nopt(to) = [1[- e [57

By substituting [57] back into [56], we obtain an
expression for Gmax(to), the maximum field attainable at a
given switching time for a class of coils having the same
design except for the number of turns.


Gmax(t0) = k I 1 e-t [58]
YR,

Gmax(to) is just the dotted line in Figure 15. The
tradeoff between switching time and field strength is
described by the plot of Gmax(to).
In summary, a design procedure has been developed for
optimizing the switching performance of a gradient coil.
Use of a current-controlled amplifier reduces switching
time. The cross-sectional area of the winding is maximized
subject to constraints that include linearity and available
space. Then the number of turns is computed from Equation
[57], given the desired switching time to. The resulting
coil will give the largest possible gradient for the desired
switching time.

Eddy Currents
Shielding efficiency of self-shielded gradient coils is
typically evaluated using a screening factor, a ratio of the
magnetic field outside the unshielded coil to the field
outside the shielded coil.63 It is possible to take this


63R. Turner, Maan. Reson. Imaa. 11, 903, 1993.









type of approach further and evaluate the ratio of the

gradient at the center of the coil with and without the

shield. This would seem to be a useful approach when

evaluating reduced-size gradient coils and comparing them to

shielded coils. For small eddy current fields, as in the

case of reduced-size coils, an iterative approximation

technique described below can be used to solve the integral

equation for the eddy current field. This technique is best

suited to situations where the eddy current field is much

smaller than applied field, so that a first-order

approximation can be used. However, it is simple and

flexible.

To estimate the eddy current field due to a gradient

coil, we assume that there is a passive shield surrounding

the coil. The shield is typically part of the cryostat. The

boundary conditions at the shield will be

(B2 B1) n = 0 [59]

SX (H2 H1) = K [60]

where B1 and H1 are the magnetic induction and field inside

the shield, B2 and H2 outside the shield, K is the surface

current on the shield, and n is an outwardly directed unit

vector normal to the surface of the shield.64 We assume the

shield is perfectly conducting, so that with H2 = 0,

H1 x n = K, [61]



64J. D. Jackson, Classical Electrodynamics, p. 1.5, John Wiley & Sons,
New York, 1975.









or more conveniently,
1
K = -B x p [62]
go
since B = o0H and n = p at the cylinder. Recall the

Biot-Savart law:

B(x) = J(x') x d3x'. [63]
4r Ix x'13

Let BO(x) be the free-space field from the gradient coil.

Then

B(x) = B0(x) + K(x') X dx [64]
4 x x'13

and substituting Equation [62] for the surface current, for

9 = Io,
1 X X'
B(x) = Bo(x) + -- B(x') x p'] x d x [65]
4 |ix x'3

where p' = p(x'). This is an integral equation for B. We

can solve it iteratively. If we define Bn(x) as the field

to nth order, then the first-order solution is

B1(x) = Bo(x) + f[Bo(x') x p'] x d x. [66]
4x Ix x'|3

The first-order solution does not take into account

eddy currents induced by eddy currents. When the coil and

shield are in close proximity, not only are the eddy

currents larger but they are also closer to the coil, so the

second-order effect can be important. To second-order,

1 x -X d2,
B2(x) = BO(x) + [Bl(x) x '] d2x. [67]
4t Ix X'1










The expressions to first- and second-order for the eddy

current field will be used to evaluate numerically the eddy

current field of several coil designs. Although the result

is not exact, the expressions can easily be integrated for

coils and shields of totally arbitrary shape, assuming they

are not too close together.

The first-order calculated eddy current field of a

68.70/21.30 radial gradient coil is plotted in Figure 17.

The first-order approximation breaks down for ratios of

shield-to-coil radius of less than about 1.5.




shield radius / coil radius

1.5 2 2. 3.5 4


-20



-40



-60
eddy
current
field
(%)


Figure 17. Eddy current field of 68.70/21.30 double-saddle
radial gradient coil. The field as a percentage of applied
field is plotted against the ratio of shield radius to coil
radius.









Coil Projects

Coil projects were intended to meet experimental needs

while exploring some aspect of coil design. The 15 cm, 9 cm

and 16 mm NMR microscopy coils are well separated from any

sources of eddy currents, and demonstrate the results that

can be achieved with simple filamentary designs and without

shielding. The CRP coil development was begun to produce a

coil with good axial linearity for NMR microscopy, so that

long, narrow samples could be observed. It seemed to be

well-suited for use as a head coil for echo planar imaging,

and we turned the development toward that possible

application.


Amplifiers

Three Techron 7540 dual-channel amplifier units (Crown

International, Elkhorn, Illinois) are used to drive the

three-axis gradient coil sets. Each axis of the gradient

coil set is split into two halves, and one channel of each

amplifier unit is wired to each half. The plane in which

the field is always zero can be shifted slightly by varying

the relative gain in the amplifiers. This is particularly

useful in the 51 mm, 7 T magnet, since the sample is

inaccessible once loaded, and mechanical centering is

difficult. The amplifiers are rated to produce 23.8 A at 42

V direct current output. The maximum slew rate is 16 V/gs.

The output impedance is less than 7 mnQ in series with less

than 3 gH, which is negligible. The power response into a 4






73


0 load is +/- 1 dB up to 25 kHz for 265 W. The noise is

rated to be 112 dB below the maximum output from 20 Hz to 20

kHz.65

Tests of the Techron 7540 were conducted into six

loads consisting of wire-wound resistors between 1 and 9 Q.

The amplifiers were pulsed to saturation at low duty cycle.

10-90% rise times were between 4 and 6.5 9s, and so are

essentially independent of load. Thus the amplifier was

bandwidth limited, not slew-rate limited, and it is

appropriate to use a linear model. The voltage and current



60 16 T
50 14
12
40 1
4 U 10
o 30 U 8
20 6 4
> H
4
10 2
0 I I 0
0 2 4 6 8 10 0 2 4 6 8 10
R (ohms) R (ohms)


(a) (b)

Figure 18. Output of Techron 7540 measured into load. a)
Measured voltage; b) Calculated current.


produced are shown in Figure 18. For load resistance of

four ohms or more, the amplifier at saturation can be

modeled by a 56 V voltage source. For a load resistance of


65Crown International, Techron 7540, Elkhorn, Illinois.









four ohms or less, it can be modeled as a 15 A current

source.

The Techron amplifiers are equipped with optional

current-control modules. With current control switched on,

an amplifier behaves like a voltage-controlled current

source. With current control switched off, an amplifier

behaves like a voltage-controlled voltage source. Current

control serves two functions when driving gradient coils.

It compensates for any variation in temperature of the

gradient coil due to resistive heating. More importantly,

it enables the coil to be switched to low fields much more

rapidly than the coil's time constant would otherwise allow.

The current-control module compares the demand (or input

signal) to the voltage across a small shunt resistor in the

output circuit. With a highly inductive load such as a

gradient coil, at high frequencies the amplifier's output

voltage is shifted almost n/2 with respect to the coil

current, and the amplifier is unstable and will oscillate.

The voltage and current response of one of the Techron

amplifiers in current mode is shown in Figure 19. The

controlled voltage overshoot reduces the current switching

time. Approximately 5 A is being switched into a 7 Q load.

An adjustable resistor-capacitor (RC) network in parallel

with the coil rolls off the high frequency gain to

compensate for the instability. The values of the RC

network are determined by the inductance of the coil. Since

the 7540 amplifiers are used with more than one coil, the









current-control units were modified so the RC networks can

be plugged in and out when gradient coils are changed.


time


1.79 ms


Figure 19. Output voltage and current of Techron 7540
amplifier with current-control module. The load is the
highly inductive 9 cm field gradient coil.



The amplifier rack was equipped with wheels and shared

between the NMR microscopy and small-animal spectrometers.

It was used in voltage-control mode with the NMR microscopy

system, and current-control mode with the small animal

system where the coil inductance was much higher. Input and

output connectors were standardized to facilitate quick

conversion. A fully-shielded output cable terminated in a

fuse-and-filter chassis eliminated interference from the RF

coils.









16 mm Coil for NMR Microscopy

The 16 mm gradient coil was developed as part of the

NMR microscope development project described below. Earlier

NMR microscopy gradient coils described in the literature

were located outside of the RF probe insert, as part of the

shim coil set. A simple and straightforward approach to

improving the coil switching time, increasing the field

strength, and decreasing the eddy current field is to

integrate the gradient coils with the RF probe. This also

allows the use of a narrowbore (51 mm) magnet. Drawbacks to

this approach include a lack of flexibility. If the

gradient coil is outside the RF probe, then any RF probe can

be used. In our approach, one gradient coil is required for

each RF probe. Also, since one of the dewars associated

with the variable-temperature (VT) control system is

replaced by an acrylic tube, the range of the VT system is

reduced. Our probe did not contain any VT control

capability. The fact that the coil former was so small

encouraged us to choose a simple design to ease the

assembly.

Since the sample-tube inner diameter was 4.5 mm and the

first metal tube, or shield, had an inner diameter of 33 mm,

this was a favorable case for using a reduced-size gradient

coil. A simple 68.70/21.30 radial gradient coil as

described above has a useful volume with a diameter of about









1/3 that of the coil,66 so the gradient former was chosen to

have a diameter of about 15 mm, or 5/8". A factor of two

remains in the ratio of the coil to the shield diameter.

This results in an eddy current field for the 68.70/21.30

Golay radial gradient coil, based on Figure 17, of about 20%

of the applied field.

The NMR microscope gradient coil set is of the

conventional Maxwell and Golay design described above. It

was constructed to accommodate standard 5 mm tubes used in

analytical NMR work. The 10 turns of 36 AWG enameled magnet

wire are wound on a 5/8" nominal outer diameter acrylic tube

(15.9 mm). Using a value of 1.36 Q/m for the wire67 and a

length of 0.135 m per turn, the resistance of each side of

the coil is 1.84 Q. The coil inductance can be estimated68

to be about 8 LH. The time constant of the coil is then

about 4 Js. The current efficiency of a 68.70/21.30 Golay

radial gradient coil is 0.918/a2 G/cm-A, where a is the coil

radius,69 so the coil has a current efficiency of 14.1 G/cm-

A. A Maxwell pair has a current efficiency of 0.808/a2

G/cm-A, so the coil has a current efficiency of 15.3 G/cm-A.

The typical figure for the linear region of 1/3 the diameter

of the coil is then enough to accommodate a sample. The

coil is driven by the Techron 7540 amplifier set. The coil



66F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984.
67D. Lide,(Ed.), CRC Handbook of Chemistry and Physics, 51st Edition,
CRC Press, Boca Raton, 1970, p. 15-29.
68F. E. Terman, Radio Engineers' Handbook, McGraw-Hill, New York, 1943.
69F. Romeo and D.I. Hoult, Maan. Reson. Med. 1, 44, 1984.









has a small time constant, so using the Techron in voltage

mode does not limit the switching time. The two halves of

each gradient coil are driven separately. Since the voltage

gain of the amplifiers can be adjusted manually, it is

convenient to vary the relative gain in the coils to shift

the zero point of the magnetic field to make up for sample

misregistration.

The details of the coil construction are visible in

Figure 47 in the following chapter. The radial coils were

wound on a flat winding former, then removed and attached to

the acrylic tube with epoxy. To eliminate any solder

connectors within the coil, the winding former allowed two

loops to be wound at once, held apart at the correct

distance. General Electric #7031 varnish was used to hold

the wires together while the coil was being clamped to the

former. No attempt was made to arrange the wire into a

packed structure. The Maxwell pair was wound around the

radial coils. The whole assembly was potted in epoxy to

secure the coils to the former, and the 36 AWG wires were

run down to a small printed circuit board mounted to the

structure of the probe. It was necessary to pot the fine

wires to keep them from moving in the magnetic field when a

current pulse is applied.

An example of the results obtained with the coil is

reproduced in Figure 60. Although the coil is capable of

about 150 G/cm, in routine operation, the coil was operated

at a full-scale field gradient of 5 G/cm for the radial









gradients, and 10 G/cm for the axial gradient, to allow

sufficient resolution.


9 cm Coil for Small Animals

An NMR magnet is frequently used for samples or animals

significantly smaller than the available bore size. It is

possible to take advantage of this fact and scale the size

of the gradient coil to match the size of the sample. One

advantage that accrues is reduced eddy current fields, since

the coil and the source of eddy currents are better

separated. Another is the increased efficiency possible

with smaller coils, since efficiency scales as the fifth

power of the diameter.70 Many applications require more

rapidly switched and more intense gradient fields than are

generally available. Diffusion-weighted imaging and

localized spectroscopy are two examples. Also, to achieve

the same bandwidth per pixel, small samples require larger

gradient fields.

The 31 cm 2 T small-animal imaging spectrometer was

supplied with a gradient coil set manufactured by Oxford

instruments that has a clear bore of 22.5 cm, and is capable

of producing a maximum gradient of 2 G/cm with a switching

time of 1 ms. Although rat, mouse, and lizard studies, do

not require the full 22.5 cm bore, they benefit from the

horizontal orientation and will not fit into other available

magnets. Additionally, localization techniques such as


70R. Turner, Macn. Reson. Imaq. 11, 903, 1993.









selective Fourier transform71 typically require better

gradient performance than is available with a large,

unshielded gradient coil set.

To meet some of these needs, a conventional Maxwell and

68.70/21.30 Golay radial gradient coil set was designed and

constructed with a clear bore of 8.3 cm in diameter. The

useful region is a sphere of about 1/3 the diameter of the

coil, or about 3 cm. The coil was designed to accommodate

rats up to 150 g, and was able to achieve 12 G/cm with a 200

ps switching time. The intense gradients are needed for

imaging experiments on smaller samples. The coil was used

for projects involving lizards, and for development of

techniques to produce diffusion images of the spinal cord of

a rat model.

We can consider the application of the time-domain

model to the 9 cm coil. As wound, the coil will produce the

field shown in Figure 20 when driven to saturation. The

Maxwell pair, Z-axis coil, is the most efficient, followed

by the inner radial, or X-axis coil, which has better

performance than the outer radial, or Y-axis coil, because

of its smaller radius. The Maxwell pair reaches the 16 A

current limit of the amplifier, and does not increase in

field beyond that point. The radial gradient coils never

reach the current limit.


71T. H. Mareci and H. R. Brooker, J. Maan. Reson. 57, 157, 1984.









G(t) (G/cm)


-f t (ms)
0.2 0.4 0.6 0.8 1

Figure 20. The gradient produced by the 9 cm gradient coil
set following a demand that saturates the amplifier.


Figure 21 describes the maximum field Gmax(to) possible

for switching time to for each of the three coils. The


Gx (G/cm)

25

20

15

10

5


I .. .' to (m s )
0.2 0.4 0.6 0.8 1

Figure 21. The maximum gradient that could be achieved by a
coil identical to the 9 cm coil, with the same cross-
sectional area, but with varying number of turns.



inherently lower inductance and resistance of the Maxwell

pair are reflected in its greater field. The actual and

optimal fields obtainable at 200 ps are compared in Table 1.

The Maxwell pair has about the optimal number of turns, and

its field is about the same as the optimum. The Golay coils









have about twice as many turns as optimum, and yield fields

about 80% of optimum level.


Table 1. Gradient fields for 9 cm coil set.

Gradient Actual Gradient Optimal Gradient
channel no. of after 200 no. of after 200
turns ps (G/cm) turns Rs (G/cm)


X 52 15.2 27.9 19.0

Y 52 12.3 26.9 15.9

Z 52 24.0 53.6 24.8


The radial and axial gradient coils consist of 52 turns

of AWG 27 enameled magnet wire. The wire was wound in a 7-

6-7-... close-pack configuration to minimize the cross-

sectional area of the winding, which is reduced by a factor

of 0.866 from a square winding pattern. The resulting

winding cross-section is about 2.6 mm on a side, only about

6% of the coil radius, so the winding approximates a

filament. The mean radius of the coils is 4.6 cm, 4.8 cm,

and 5.1 cm. The Maxwell pair is wound on the outside

because it is inherently more efficient and will hold down

the other coils. The two halves of each coil are wound

separately, and one channel of a stereo amplifier is wired

to each. It is driven in current mode from the Techron 7540

amplifiers. The current-control circuit helps to buck the

inductance of the coil. The coil resistance for the radial

windings is predicted to be about 6.7 Q for the inner set.

The measured resistances and time constants including the









leads and filters are given in Table 2. The time constant

of the shorted power cable with filters and fuses was too

short to measure with the amplifier, so it can be assumed to

be negligible. The last column is the inductance estimated

from the Bowtell and Mansfield formulation for coils on the

surface of cylinders.72 To adapt for the thick winding, the

height is added to the width of the winding. Calculations

for loops of square cross section compare closely to

heuristic formulas.73


Table
for 9

Coil




X1

X2

Y1

Y2

Zl

Z2


2. Comparison of measured and predicted inductance
cm gradient coil set.

Measured Measured Time Experimental Theoretica
Resistance Constant Inductance Inductanc
(Q) (9s) (gH) (UH)


7.02

7.08

7.57

7.61

3.50

3.53


175

175

195

190

150

150


1229

1239

1476

1446

525

530


l
e


1727

1727

1866

1866

330

330


The difference between the theoretical and experimental

inductance is not due to the inductance of the power cable,

which was measured by the same technique to be about 18 pH.

It is primarily due to poor control of the cross-sectional



72R. Bowtell and P. Mansfield, Meas. Sci. Technol. 1, 431, 1990.
73F. E. Terman, Radio Engineers' Handbook, McGraw-Hill, New York, 1943.









dimensions of the winding, which expands when it is removed

from the winding former. The maximum operating field as the

system has been installed is 12.83 G/cm for X, 12.28 G/cm

for Y and 8.80 G/cm for Z. By limiting the gradient field

to a value below that corresponding to the steady state

current, the switching time is better controlled.

A drawing of the coil that illustrates how the Faraday

shield and RF coil fit together is shown in Figure 22.


Golay windings











RF coil Faraday Centering Maxwell Clamping
shield disk pair cam

Figure 22. Drawing of 9 cm gradient coil set with Faraday
shield and RF coil.


Figure 23 is a photograph of the assembly. The former

is an acrylic tube of 3.5" (89 mm) nominal outer diameter

and 1/8" (3.2 mm) wall thickness. The radial gradient coils

are wound on rectangular bobbins made from three flat pieces

of acrylic. The wires are held together in a hexagonal

matrix by General Electric varnish #7031 diluted in acetone.

After winding, the bobbin is disassembled and the coil is

glued onto the cylindrical former with epoxy. A variety of
































Figure 23. Photograph of 9 cm gradient coil set. The power
cable and water supply cables are visible at left. The
axial and radial gradient coils are visible through the
cooling tubing.


RF coils were developed as inserts for the probe, including

19F, 1H birdcage, 31P/1H double-tuned saddle coil, and my

own 1H saddle coil. A pair of cams connected by a rod and

mounted on the edge of the mounting flanges served to lock

the probe into the magnet. A Faraday shield was used in

addition to the filter/fuse box to isolate the RF coils from

the gradient coils. The shield consisted of strips of

Reynolds heavy-duty aluminum foil approximately 2" wide,

overlapped by about 1/2", and insulated from the other

strips by masking tape that also secured the strips to

manila card stock. At one end, all strips contacted a

header strip. Provision was made to ground the shield, but









in practice it was not used. The interdigitated geometry

reduced eddy currents from the gradient coil, but allowed

the shield to serve as a barrier to the RF field. The

Maxwell pair is on the outside, which helps to hold the

radial gradient coils in place. Epoxy was initially used to

hold the windings to the former and pot the windings, but

the epoxy did not withstand the temperatures developed by

the coil and depolymerized. Polyester was selected as a

casting resin that would outperform the acrylic former under

warm conditions, and the coils were potted in polyester.

In order to cool the unit, approximately 25 m of 1/8"

O. D. by 1/64" wall polypropylene tubing was wound around

the coils. It was connected to a Neslab circulating system

that is capable of producing a pressure head of 40 psi.

Supply tubing consisted of about 30 m of 1/4" 0. D. by

1/32" wall polypropylene. Assuming laminar flow, the water

flux through a tube74 of radius r (cm) is

flux = r 4Ap
8 l

where Ap is the pressure in dyne/cm2, 1 is the length of the

tube in cm, and g is the viscosity in poise. The resulting

flux for a 40 psi drop is 8.6 cm3/s. One KW of power

transferred to the water in the tube will raise its

temperature by about 280 C. Measurement of the motion of

bubbles in the tubing reveals a flow of about 3.5 cm3/s.


74D. Lide,(Ed.), CRC Handbook of Chemistry and Physics, 51st Edition,
CRC Press, Boca Raton, 1970, p. F-34.









The difference is almost certainly due to quick-release

connectors that allow the probe to be removed or inserted at

operating pressure.


15 cm Coil for Small Animals

The 15 cm coil was designed to accommodate larger rats

and other medium-sized laboratory animals and still produce

a higher field and faster switching time than the Oxford

gradient set. Like the 9 cm and the NMR microscope coil

set, it is based on filamentary winding design. Since it is

also driven by the Techron 7540 amplifier set, which is

under-powered for a coil of this diameter, switching

performance was at a premium. So, in contrast to the 9 cm

coil, the 15 cm coil was designed to have optimal switching

performance for the chosen switching time. In order to

provide more flexibility in choosing either high field

intensity or fast switching time, the windings were split

and could be driven either in series or parallel. Since the

coil would tend to become somewhat unwieldy as an insertable

unit, its length was the shortest that would give

essentially undiminished field intensity. Plots of the

relative error in Figures 26 and 27 illustrate the linear

region of the radial coil design. In order to avoid

aliasing signals from long animals, the axial coil was based

on an extended linearity design by Suits and Wilken.75 The


75B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.









coil and amplifiers were capable of developing 9 G/cm in a

100 gs switching time on X, Y and Z channels.

The arcs in the 15 cm coil were arranged to have the

minimum length without losing a significant amount of

efficiency in the static limit. The standard solution of

68.70/21.30 for the arc positions arcl/arc2 leads to no

third order component from either arc, but a family of

solutions for which the third order components cancel is

available. These solutions are graphed in Figure 24.


arc2 (deg)


60

50

40

30

arcl (deg)
30 40 50 60

Figure 24. The solutions to the arc position of the double-
saddle radial gradient coil.


Each solution is graphed twice, since exchanging arcl and

arc2 results in the same coil design. To improve the

relative size of the linear region to the coil, one would

like to make the coil shorter than the 68.70/21.30 solution.

The current efficiency decreases with the length, since the

return arcs tend to cancel the desired field, and moving

them closer increases the effect. However, in order to










include the effect of reducing resistive loss in the coil,

one must divide the current efficiency by the square root of

the length. The resulting measure is an indication of the

relative field that can be produced with an amplifier of a

given power. Figure 25 illustrates the relative power

efficiency as a function of the position of the return arc.


rel. power eff.

0.3

0.25

0.2
0.15

0.1

0.05
angle (deg)
20 25 30 35 40 45 50

Figure 25. The relative power efficiency of the double-
saddle radial gradient coil as a function of the angle
between the z-axis and the current return path.


The peak efficiency is achieved at 260, but efficiency is a

weak function of angle and much shorter coils can be used

with little loss of performance. Based on this curve, the

arcs of the 15 cm radial coil were placed at 30.20 and 66.10

from the axis of the coil, compared to 21.30 and 68.70 for

the Golay coil. The overall length of the coil is reduced

by 33%. The current efficiency is 0.819/a2 G/cm-A compared

to 0.808/a2 G/cm-A for the 68.70/21.30 Golay coil. It would

be even better to use a variant of the field-versus-

switching-time approach to determine the change in

















0.4

-1%
-3%
0.2
-1%



Z 0 +1%+3%+5%



-0.2




-0.4





-0.6 -0.4 -0.2 0 0.2 0.4 0.6

a) Y







0.4 +5%

+3%



+1%
0.2






Y 0






-0.2





-0.4

-0.4 -0.2 0 0.2 0.4

b) x




Figure 26. Relative error plots of 30.20/66.10 radial
gradient coil. Radius of coil corresponds to 1 on scale.
a) YZ plane; b) XY plane.


1












0.61
+5%
+3%
0.4 +1%
-1%
-5%
-3%
0.2


Z 0 +1%+3%+5%


-0.2


-0.4


-0.6
-1 -0.5 0 0.5 1


X




Figure 26--continued. Relative error plots of 30.20/66.10
radial gradient coil. Radius of coil corresponds to 1 on
scale. c) XZ plane.



performance for coils of different lengths, taking the

changing inductance into account.

The size and shape of the linear region of the 66.10

/30.20 radial gradient coil design is described by the plots

of Figures 26 and 27. It is not greatly different from the

longer 68.70/21.30 design. Note from Figure 26 that the

regions of equal relative error are not simply connected. A

three-dimensional plot of a region as large as that in the

two dimensional plots would give a false impression of the

size of the linear region, since the apparently solid

volumes would contain large holes. To avoid these "bubbles"

of linearity and give a true picture of the useful volume,


r










the region displayed in Figure 27 is truncated. As a result

of the truncation, it is possible to see through the linear

region. All plots were produced by direct evaluation of the

Biot-Savart law.


-2

z
0

1





0.5

x 0


-0.5


-1
0.5
0
-0.
Y -1


Figure 27. Perspective rendering of the 5% relative error
region of 30.20/66.10 radial gradient coil.


The axial coil was built to an extended-linearity

design proposed by Suits and Wilken.76 It consists of loops

at both 40.00 and 66.30 from the z axis. The outer loops

carry 7.5 times more current than the inner loops. Using

the additional degrees of freedom of the second-loop

position and the ratio of current in the loops, the fifth


76B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989.









and seventh order terms are canceled, resulting in about

eight times the useful volume of a Maxwell pair. For a coil

with more turns in some loops than others, the current

efficiency does not have an unambiguous definition. With

respect to the current in the outer loops, the current

efficiency of the coil is 0.635/a2 G/cm-A.

The 15 cm coil was matched to the Techron 7540

amplifiers in our laboratory using the time-domain model of

gradient performance described above. The height and width

of the windings was set to be 1 by 1 cm for the radial and 1

by 2 cm for the outer axial, to ensure that the assumption

of filamentary wires in the calculation of angle position

would be valid. The width of the outer axial winding was

increased from 1 to 2 cm, since it is farther from the

center than the others, and it was necessary to increase it

to match the radial performance. Inductance of the radial

and axial coils was calculated using a Fourier-Bessel

approach.77 The available combinations of switching time

and maximum gradient are shown in Figure 28.

In order to improve the switching time to smaller

fields, the 15 cm coil is constructed from split windings.

All the gradient coils described have the two sides driven

by separate amplifiers so that the magnetic center of the

coil can be moved. The 15 cm coil has each side split

further into two identical but electrically separate



77R. Bowtell and P. Mansfield, Meas. Sci. Technol. 1, 431, 1990.










windings. Placing the windings in parallel trades field

intensity for switching time; placing them in series reduces

switching time at the expense of lower field intensity.


Gma (G/cm)

14 X
12
10
8
6
4
2

to (ms)
0.5 1 1.5 2

Figure 28. Maximum field Gmax (G/cm) vs. switching time to
(ms) for the 15 cm field gradient coil set as driven by the
Techron 7540. Points along the curves represent designs
with different numbers of turns, increasing from left to
right. The top curve represents the inner radial coil. The
lower curves represent the outer radial coil and the axial
coil. A "slower" coil has a larger maximum field.


All three coils are optimized to approximately 10 G/cm

for the series mode. The exact number of turns for the

desired field is not used, due to the limited number of

standard wire sizes and the use of rectangular winding cross

sections. For the number of turns actually used, to and

Gmax based on the time domain model are tabulated in Table

3.

A photograph of the 15 cm coil assembly is shown in

Figure 29. The coil is wound on an acrylic former having a

nominal O. D. of 6" (152 mm) and a wall thickness of 1/4"

(6.4 mm). The overall length was 38 mm. The radial coils




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